The difference is that the same note can be played in different octaves. As Dom answered, the technical meaning of an octave is that the frequency of the pitch doubles. Practically speaking, to most people these notes sound like the same note, only one is higher.
On the violin, you can play three different G's in first position (where a beginner usually plays). The lowest is the open G string. The next is played with the 3rd finger on the D string. If you can bow two notes at once, try playing both the open G string and the G on the D string. If you play it perfectly, it will be a very smooth sound, almost as though there was only one note being played. You can contrast that with how it sounds when you play both the open G and any other note on the D string at the same time. If you can't bow two notes and once and don't have someone to demonstrate it, you can try plucking the two notes at the same time, but the similarity is not as easy to hear that way.
The third G is played with a low 2nd finger on the E string, so is a little more difficult to find than the other two, and cannot be played with the open G string to compare.