PBS Kids has lots of games and videos for kids.
Some games are aimed at very young kids—e.g., in Bubble Pop the child clicks on bubbles that rise out of Curious George's bathtub and counts them, 1, 2, 3, . . ., while Curious George giggles. Others for little kids help teach colors, shapes, etc.
Some games are aimed at older kids. For example, Cyberchase games are designed for students who are 8-12 years old.
How do you find an appropriate game for the skills your student is working on? Take a look at PBS Teachers. A navigation bar lets you choose a grade level (Pre-K, K-2, 3-5, 6-8, 9-12). Then you can choose a subject (The Arts, Health & Fitness, Math, Reading & Language Arts, etc.). The result is a long list of videos, interactive games, and offline activities. You can search within the list.
If you're looking for math games, go to the Parent & Teachers page for Cyberchase.
And for a variety of science, technology, and math lesson plans and resources, for K-12, see the STEM Education Resource Center
A lot of the games require Flash, so will have to be played on a computer.
If a tutor likes to let a student play a game on an iPhone or other mobile device, use the mobile version of PBS Kids (m.pbskids.org).
The PBS sites for parents and teachers have some ads (the ones I saw were for the Seattle Aquarium and Applegate Organic & Natural Meat). The PBSKids site is free of ads—something I look for in sites for students.
Kids are exposed to plenty of advertising as it is. Why distract them with ads for, say, KeyBank, Target, Amazon, and Marriott (the ads I saw on Learning Games for Kids.com when I visited it just now)? I make an exception for Free Rice.com because the sponsors are supporting world hunger relief. This isn't to say I'd never recommend Learning Games for Kids.com—or other sites with ads. I just prefer not to have ads.