What's wrong with my computer? Graphics card?

I am a gamer, I use my computer mainly to play video games. Which also is called a "gaming laptop" by it's description when I bought it. Only thing is, it can't run certain games. It has a lag and I've pressured my isp and they say it's not the internet. But anyway, it's mainly lag but in some points the entire computer computer just shuts down. The only thing I can really think of is the graphics card...please help...I do not enjoy having my entire computer shut down on me when I'm playing..especially on games such as age of empires..