What is it that you don't understand?
In simple words, what the meaning of this passage:
"fundemental groups is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other"
When would it not be possible to deform one path in a space, say the plane R^2, into another path? Answer: never. Any path can be deformed into any other, just by linearly interpolating corresponding points.
But remove the origin from the plane, and consider two circular paths, one whose interior contains a neighborhood of the missing origin, and another which doesn't go around it. Can you ever deform one into the other? No--you can't move a path across a missing point continuously.
On a torus, a path that goes through the hole can't be deformed ("homotoped" is the word) into a path that goes the long way around the hole. And neither one can be homotoped into a path that doesn't go around or through the hole.
If two paths have the same base point, you can concatenate them to get a new path with the same base point. This is the group operation. The structure of the resulting group gives you information about the topological space.