These days my gaming is pretty much exclusively on my 4th gen iPod Touch, so all games mentioned are iOS, but most (I think) have an Android version too.
EDIT: Have now got an iPad mini Retina, so some of these I'd only recommend for newer iDevices. Those that work well/acceptably on the last gen iTouch are tagged with (iOS 6).
Stuff I'd recommend:
- The Walking Dead - see impressions thread
- Bastion - see impressions thread
- Galaxy on Fire 2 (iOS 6) - cracking spaceship shooting/equipping goodness - currently the nearest you'll get to Elite on a mobile. If you've got an iPad get the HD version.
- Swordigo (iOS 6) - a pretty darn fine platform action adventure.
- Tiny Troopers (1 + 2) (iOS 6, although it struggles a bit) - it's basically Cannon Fodder, but on your mobile
- Infinity Blade (iOS 6) - 1-on-1 hack and slash fun
- Tron (iOS 6) - sadly the lightbike controls are rubbish and annoying (well, I couldn't get on with them) but the tank game is good enough to be worth a mention, and it's free
- Hoplite (iOS 6) - see impressions thread
- Greed Corp (iOS 6) - a rather different take on the TBS genre. iTouch is a bit small to do it justice, but this'd be cracking on an iPad (if you get the HD version).
- Hunters: Episode One (iOS 6) - awesome TBS that's pretty much a mobile version of Space Crusade / Space Hulk
- Hitman GO - see impressions thread
- Plasma Sky (iOS 6) - awesome retro-styled bullet nightmare shooty fun. See impressions thread
- PewPew (iOS 6) - old school beat-the-score manic shooting goodness, with 4 different game modes.
- Sonic Jump (iOS 6) - not the classic Sonic style, but a well presented platformer that I keep returning to.
- Peggle (iOS 6) - puzzle and skill goodness
- Cut the Rope (iOS 6) - a puzzle classic
- Where's My Water? + Where's My Perry? (iOS 6) - physics based puzzling at its best
EDIT 11/9/2014: Added link to Plasma Sky impressions.
EDIT 18/9/2014: Added Bastion and iOS 6 tags.