The Observable Universe
When we talk about our Universe, we make a distinction between "the Universe" and "the Observable Universe". The latter includes only what we can see. By "can see", I don't mean what we have the technology to detect. Rather, I mean all objects out there from which light has had time to reach us given the age of the Universe, the speed of light, and the history and future of the expansion of the Universe. The age of the Unvierse is 13.8 billion years. Because the speed of light is finite, we can't see anything that is so far away that light would have taken longer than that from us to reach us. This isn't a technological limitation; this is a limitation on whether or not there is light, even in principle, for us to see given as much technological prowess as you could want.
Indeed, as we look towards the edge of the Observable Universe, we're looking back in time. If light took us 13.7 billion years to reach us, then we're seeing the Universe out there as it was 13.7 billion years ago, not as it is now.
The Universe as a whole, however, is probably infinite. This is easy enough to say, but it's a rather difficult concept to wrap your brain around when you really start thinking about it. One solution is not to think too hard about it. If you find yourself asking questions like "if it's already infinite, how can it expand?", you're not thinking properly about infinity. Infinity is a concept, not a number.
However, the Universe doesn't have to be infinite. According to General Relativity, there are other possibilities. I'm going to lump those possibilities into two categories, but only really talk about the latter.
It's possible that our Universe has an interesting topology. Topology is different from geometry. Geometry includes things like lengths of lines, radii of curvature, sums of angles inside polygons, and so forth. Topology talks about how different parts of space are connected.
As an example, consider the classic video game asteroids:
This game takes place in a (very small) two-dimensional universe. The geometry of the Asteroids Universe is Euclidean— that is, parallel lines will never cross, the ratio of the circumference to the diameter of a circle is π, the sum of the three interior angles of a triangle is 180°, and so forth. However, if you ever played this game, you know that if you go off of the left of the screen, you come back on the right side of the screen. Likewise, if you go off of the top of the screen, you come back on the top. This universe is unbounded; you never hit a boundary, or an edge. However, it is also finite. Its topology is toroidal; it has the same topology as the surface doughnut, although it does not have the same geometry of a doughnut. (The surface of a doughnut has curvature.)
It's possible that our Universe is similar. It may have a flat geometry, but a topology that means that if you kept going in one direction, you came back where you started. If it does have this topology, it's on spatial scales larger than the Observable Universe. Otherwise, we would have seen the signature of this topology (i.e., the fact that parts of space are effectively repeats of each other if you keep going far enough in one direction) on the Cosmic Microwave Background.
For the rest of this post, I shall assume that the Universe does not have any interesting topologies like this. Either it's just infinite space, or it's finite space that is the 3d equivalent of the surface of a sphere.
Possible Geometries of the Universe
The geometry of our Universe doesn't have to be Euclidean. Depending on the total energy density (including the density of regular matter, dark matter, and dark energy), there are three possibilities for the curvature of our Universe.
The parameter Ω is a convenient way of talking about the density of the Universe. There is a critical density, which depends on the current expansion rate of the Universe. That critical density is about 9×10-30 g/cm. That doesn't sound like a lot, but remember that the Universe is mostly empty space! Where we live, on Earth, is an extremely high-density place compared to most of the Universe. The parameter Ω is defined as the ratio of the density of the Universe to the critical density. If Ω=1, then the Universe has a flat geometry. Note that "flat" here doesn't mean "two-dimensional", the way you may used to be talking about flat. Rather, it means that the geometry of space is Euclidean, just like the geometry you probably learned about in high school.
On the other hand, if Ω>1, the Universe has a "closed" geometry. In this case, the geometry of the Universe is the same as the three-dimensional surface of a four-dimensional hypersphere. If that sounds like gobbledygook, think of it as the 3d equivalent to the surface of a sphere. Note that there doesn't need to really be a fourth spatial dimension or a 4d hypersphere out there. It's just that the geometry of the Universe— how parallel lines will behave when extended, how the angles of triangles will add up, what the ratio of the circumference to the diameter of a circle will be— is the same as the geometry of the surface of a sphere. It's possible to describe the mathematics of this geometry entirely using only three spatial dimensions, so there's no need for a higher spatial dimension in which to embed our Universe. However, for purposes of our visualization, it's worth thinking about the surface of a sphere, as that helps us get some idea about what sorts of things would be true in such a universe. The surface of a sphere is a two-dimensional closed universe. Remember, that the universe is the surface. There is no center to this universe, not within the universe— for everything within the universe is on the surface of the sphere, and no point there is any different from any other point.
If Ω<1, the Universe has an "open" geometry. This is harder to visualize. It turns out that you can't embed a slice of an open 3d universe into three dimensions to visualize it, the way you can with a closed universe (in which case you get the surface of a sphere, as described above). However, the closest two-dimensional equivalent would be a saddle or a potato chip (each of which is a hyperboloid or hyperbolic paraboloid). This is an unbounded and infinite universe. It keeps on going forever. However, it's also clearly not flat, and so will have an interesting geomoetry.
The Geometry of Our Universe
You can figure out the geometry of your universe several ways. One way is to create a triangle by drawing three straight lines through space. Then, measure the angle between each of those pairs of lines. If the three interior angles add up to 180°, you're in a flat universe. If they're more than 180°, then you're in a closed universe; if they're less than 180°, you're in an open universe. The problem is the precision needed for this measurement. In order to be able to tell whether or not the angles add up to 180°, you either need to measure them mind-bogglingly precisely, or you need to draw huge triangles, such that the length of one side of the triangle approaches the radius of curvature of your universe. (How close it approaches it depends on how precisely you can measure angles.)
Effectively, we have done this. Measurements of the Cosmic Microwave Background (CMB) give us triangles. One leg of the triangle is given by the characteristic size of fluctuations in the CMB. We know the physical size of those fluctuations. The other legs of the triangle are given by the path of light travelling from either side of one of those fluctuations. By measuring the angle between the light coming from either side of a fluctuation, we can figure out what the geometry of this isosceles triangle is. We did this. The answer: our Universe is flat. However, as with all physical measurements, there is uncertainty on this measurement. The latest form of this measurement tells us that Ω must be between 0.9916 and 1.0133, to 95% confidence (see "Reference" at the end for the source of these numbers). That means that there still is the possibility that our Universe is either infinite (in the case of Ω≤1) or finite (in the case of Ω>1).
The Minimum Size of Our Universe
The Universe is big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to the Universe.
With all due apologies to Douglas Adams, let's quantify how big our Universe is.
First, the age of the Universe is 13.8 billion years. That is a long time compared to you and I, but as compared to the age of the Universe, it's just about right. The edge of the observable Universe, right now, is 48 billion light-years away. "Wait!" you may cry. "How can light from something 48 billion light-years away have reached us in a mere 13.8 billion years!" Remember that while that light was working its way towards us, the Universe was expanding. The light, in a sense, had to try to "catch up" with that expansion. This is imperfect language, and indeed if you know Special Relativity you should object to it. However, it does (sort of) make sense in the context of General Relativity.
How does this size compare to the size of the Universe as a whole? If we make the assumption that Ω=1.0133— the highest total energy density consistent with our current data, and thus the smallest closed universe consistent with our data— it's possible to calculate how big the Universe is. The result looks something like the following:
In this picture, the surface of the sphere is meant to represent the whole Ω=1.0133 Universe. The parts that are "greyed out" are outside our Observable Universe; the patch at the top that you fully see is the Observable Universe. The radius of curvature of this universe is 120 billion light years. Its circumference is 760 billion light years. That means that the diameter of our Observable Universe is just 1/8 of the full length of a line you'd have to draw through space if you wanted it to connect back go yourself. The volume of the whole Universe is about 100 times the volume of our observable Universe. (If you object to the fact that 83 is not equal to 100, remember that we're not talking Euclidean space here, so your intuition for how radii and volumes of spheres relate doesn't entirely apply.)
Remember, though, that this is the minimum size of the Universe given our current data. Most of us suspect that the Universe is really a whole hell of a lot bigger than that, and indeed may well be infinite.
Size and Fate Are Separate
If you read almost any cosmology book written more than 12 or so years ago, and some written since, you will probably read something about a closed universe being one that recollapses, and an open universe one that expands forever. This is true only if the dark energy density of the universe is zero! Implicitly, those texts assumed that our Universe was matter dominated, and as such the geometry and fate of the Universe were linked. In a universe such as ours, where there is dark energy, the fate and geometry are not so tightly linked. Dark matter and dark energy both affect both the shape of the Universe and its ultimate fate, but they affect it differently. Exactly what will happen to our Universe depends on the details of what dark energy really turns out to be. However, for what most of us consider to be the most likely versions of dark energy, the Universe will keep expanding forever, with clusters of galaxies getting ever more and more separated. This is true whether the geometry of the Universe is flat, open, or closed.
The numbers for the current expansion rate of the Universe (used to derive the critical density) and for the limits on the curvature of the Universe come from the cosmological implications of the WMAP 7-year data as described in Komatsu et al., 2011, ApJS, 192, 18. The image used to wrap the universe sphere is the Hubble Ultra-Deep Field.