It used to be that if you wanted to play even the most basic of games, integrated graphics would not suffice. That is not the case now as the major players have invested much effort into making high quality integrated graphics.
A computer has integrated graphics when the graphics controller is embedded within the motherboard as opposed to a separate, removable, card.
Advantages
-Much less expensive than graphics cards
-Improving with every new release, the major players have investments in them
-On par or faster than older graphics cards so older games will likely run just fine.
Disadvantages
-Much slower than the latest graphics cards.
-Use the system memory, taking up memory that could be used for other programs.
To play the latest games in all of their hi-res glory, you will need to have a standalone graphics card.
Advantages
-Much faster rendering and more advanced features than integrated graphics.
-Can be overclocked and modified for even more performance.
-Use their own dedicated memory so your other applications are not affected.
Disadvantages
-More expensive than integrated graphics.
-A little more difficult to buy since so many different companies offer so many different features. A lot of research is required to buy the one that is right for your needs.
So there you have it. Unless you intend to play the latest games at full-resolution, an integrated graphics controller will work just fine for you.