A couple classics:

Backgammon
Othello
Cribbage

And some newer games:
Istanbul
Dominion
Lost Cities
Patchwork
Jaipur
Hive

Star Realms
Targi
Carcassone




[More difficult but so much fun]

Castles of Burgundy
A Few Acres of Snow (bonus - learn about French-Anglo relations during early colonial days)

Thanks so much for these suggestions!

Ticket to Ride -- especially the expansions designed for 2-3 players specifically (e.g., Switzerland).

Mastermind

We play Quirkle as two player a lot - although more can play. It's like Scrabble, but with shapes and colors instead so there's no advantage based on vocabulary.

we just got Ticket to Ride Junior version and have been having so much fun playing it. My 6 and 8 year old picked it up really quickly and we've been playing it as a family. Its also made for 2 people and I am sure the adult version is more fun. you should check it out.

Rummikub and Colorku.

Colorku is Sudoku, using colored marbles instead of numbers. Although it is designed to be single player my son and I often tackle harder levels together.

Ticket to Ride!

My 10 year old loves Racko and I broke out Battleship the other day to play with her.

For board games, I recommend 10 Days In... (it's a series. there are multiple boards.)

I grew up in a two person family, and I remember playing a lot of poker, blackjack, cribbage, and rummy.

Boggle
Connect Four
Mancala

Cribbage, bananagrams

Farkle: fun dice game
Dutch Blitz: card game
Forbidden Island: cooperative game, similar to but simpler than Pandemic

Hive, Jaipur, Patchwork, Splendor. The first three are 2 player only.

We are a 2-person household, just my 10-year old DS and me. We both like to play games and would like more ideas for fun 2-person games. Board games especially welcome. We do play cards, chess, and checkers. I plan to get him Scrabble for Christmas. Are there other fun games out there especially good for 2 people?
Thanks.