Many fruits (tomatoes being one) and vegetables are picked before they are ripe and then artificially ripened at their transport destination using artificial means like ethylene gas. This makes fruits and vegetables make it to the store and last longer there without spoiling, and is the reason we have many of our vegetables year-round. The down-side to this technique is that the flavor of them is nowhere near what a garden-ripened vegetable or fruit would be.
A vine ripened tomato is one that has been allowed to ripen completely on the vine until it is at or near its peak, giving much better flavor. Because they are so perishable they tend to be much more expensive and harder to get. Supermarkets won't have them but farmers markets will depending on the season. Good restaurants pay a lot of money for a good year-round supply of quality vegetables.
I grow my own and I can tell you that there's not comparison between even the best store-bought tomatoes and the ones from a good garden. Not hard to grow either if you pick the right variety.
That being said, the term vine-ripened is often misused as a bit of marketing speak to make them sound more attractive, however if you're at a good restaurant it's probably true.