I'm going to cheat a little and post ten, five old classics and five newer games. Also, because the person you are recommending the games to matters quite a lot, I'm not going to put them in any specific order, but instead choose one from 5 of my favourite genres.
Adventure (Best genre for new players, in my opinion):
Old:
King's quest 6 (PC) A lot of people cosider it the best in the series, and it's definitely the easiest to get into. It's also pretty much the only game in the series that can be enjoyed at face value with no nostalgiagoggles required.
New:
Ghost Trick, Phantom Detective (DS) An awesome little game, one of my favourites for the ds. It's a bit heavy on the plot-to-gameplay ratio but if you don't mind reading a little, it's a must play. The clever gameplay mechanics certainly help.
Platformer/Action-platformer:
Old:
Super Mario Bros. 3 (NES/SNES/GBA...) I don't think I have to explain this one.
New:
Contra 4 (DS) I know this isn't the most beginner friendly game available, but if you don't mind a genuine challenge, you can't go wrong with contra. Contra 4 should teach the new gamer that not all new games do everything for you if you just cry a little!
Strategy (turn based, because it's easier for new players):
Old:
Civilization 2 (PC) My personal favourite of the series. 4 was a better game, but for some reason I still prefer this one (Can't say anything about 5, haven't played it). It's a classic, what more can I say.
New:
Advance Wars: Dark Conflict/Days of Ruin (DS) (A lot of ds games, I know). I didn't want to say Civ 4 so I chose this instead. Again, the best game in the series. Easy enough to get started with, but the difficulty rises pretty quikly to give a good challenge.
RPG:
Old:
Fallout 2 (PC) Turn based combat again, makes it easier to get into, since you don't need to be that familiar with the controls to react quikly. This is one of my favourite RPGs of all time. The plot is better in the first one, but Fallout 2 just has a lot more for you to do.
New:
Elder Scrolls 4: Oblivion (PC) I know some of you would probably put Morrowind here instead, but I think Oblivion is better for beginners. It even has a handy difficulty slider for those who think it's too tough. It has its problems and that's why you should get it for PC. Mods can fix a lot.
Stealth:
I cant think of an old stealth game, at least not one that is good, so I'll just give you a new one:
Hitman: Blood Money (PC) I'll have to choose the best game in the series again. Great game with many different ways to beat the levels. And the easiest difficulty has infinite saves/mission so it should be ok for beginners too.
FPS:
I don't really play modern shooters so I'll just say Doom.
EDIT: Totally forgot X-Com. Should have put that instead of civ 2.