The phrase that comes to mind here is finite but unbounded
. That is, you can travel forever through the Universe and never reach the "edge", but it's not endless.
How could this be? Well, think of the surface of a sphere, such as the surface of the Earth. It's the same kind of thing. Except the Universe has three physical dimensions rather than the two dimensions of the surface of a sphere. So it might help to think of the Universe as the three-dimensional "surface" of a four-dimensional hypersphere. The monkey mind wasn't built to think of 4D structures, so you can only do it be analogy with the 3D sphere example, but you get the general idea.
Things really get interesting when you consider that Big Bang is kind of like a Black Hole in "reverse", and that the Universe might be the "interior" of a black hole. An event horizon such as black holes are framed by would make a nice "end" to a universe without appearing to be one. The edge is no edge.
This all opens the way for the extremely compelling idea of cosmological natural selection, as sketched out by Lee Smolin in his book The Life of the Cosmos. Highly recommended.
I hope this helps.