Seattle is a great place to live or visit and has a fairly decent collection of places to play arcade games and pinball at...

Shorty's
shortydog.com
2222A Second Ave in Belltown (Downtown Seattle)
The number one destination for pinball and arcade games in Seattle. It's 21 and over and has many classic pins and vids, a handful of newer games, the machines are in good shape and played often. It's a great place to go during the afternoon or after work to play some pinball and fun at night, although in the evenings it can get quite packed. The allure of this place is almost too great. They have $2 pints of PBR (and better stuff), cheap "Coney Island" hot dogs, veggie dogs, sausages, a hidden back bar and lots of games! There are usually about 8 vids and they are rotated in and out, the classics mostly come in the way of the '80s reunion multi-game machines with a couple of modern shooting games thrown in. Shorty's highlight is their pins--a full backroom of them. This is where serious pinheads come to play and lots of regular folks get their silver ball fix.

Gameworks
gameworks.com/locations/seattle.php
1511 Seventh Ave in Downtown Seattle)
Gameworks is a big national chain. They have all the newest arcade games, including shooting, fighting and large interactive and multiplayer arcade games. They usually have a few classic reunion type of arcade games like Ms. Pac and Galaga and a few pins in one corner, but this isn't really the place for classic games, it's the place for new games and multi-players. Also includes a bar and restaurant upstairs with pub type of food.

The Sunset Bowl
sunsetbowl.com
1420 NW Market Street in Ballard (NW of Downtown)
While primarily a bowling alley, the Sunset Bowl has a good-sized arcade with a few pinball tables and a bunch of newer arcade games.

Sureshot Espresso
nwsource review
4505 University Way NE in the U-District (N of Downtown)
A few cocktail arcade games dot the main room, but the gold is in the small back room where you will always find a few older EM pinball machines (Wizard and Hokus Pocus last time I was there) and a few vids (Marble Madness and Monaco GP).

The Twilight Exit
yelp.com review
2020 E. Madison Street in the Central District (E of Downtown & Broadway)
Loungey bar with a decent kitchen for food, this is more a place to meet a friend for drinks or hit Sunday night karoke than play games, but the owners have always had quite a few arcade games and pinball machines in the bar. Granted half the video games are usually broken cocktails that are turned off, but there are always at least two pins and a few vids running.

University of Washington HUB
hub gameroom website
UW Campus in University District (N of Downtown)
I played games here constantly in college, I always held the highscore on Atari Assault and some of the pins. Today the game selection is a bit sad, the emphasis in the Hub game room is on the pool tables and bowling alley. But there still are a couple pins and quite a few vids (although the game list on their website isn't up to date.

Dante's
yelp.com review
5300 Roosevelt Way NE in University District (N of Downtown)
At night this sports/college bar is packed to the gills for it's cheap pitchers, loud music, games and party atmosphere. Suprising is the emphasis on games here though, they have pool tables, air hockey, 2-3 pinball tables, foosball and a bunch of newer arcade games. I kinda want to hate this place, but with 4+ people that are drinking it's always fun. Watch out for their jello shots, they come in hypodermic needles and will kick your ass!

Seattle Center Fun Forest
funforest.com
Seattle Center in Downtown Seattle
When I was a kid the arcade and pinball game collection at the Fun Forest was huge and legendary. In fact if you follow the link on the main page to the old EM and game collection photos you will see 100 games that were there in the early '80s. Today it pales in comparison, but there are still a bunch of newer arcade games, generally 4-5 pinball tables, and larger interactive pins. Plus, it is all-ages.

Jules Maes
julesmaessaloon.com
5919 Airport Way South in Georgetown (S of Downtown)
South of downtown about 15 minutes off I-5 is Georgetown. Jules Maes was bought and restored by the owners of Shorty's, so it should be no surprise that is has 5-6 pinball tables and a bunch of arcade games. While it's emphasis is more on being a bar and a back room with live music, this is still and excellent place to play games at.

There are definitely quite a few bars and other places around town that have a few games in the corner, so keep your eyes open! Seattlepinball.net usually has the most up to date listings of pinball table locations around town if you want to play a specific game.