Fortress Legends is an enjoyable little dungeon romper, similar to The Quest for Epic Loot. It's essentially f2p (I haven't payed anything so far). By Namco Bandai, it's currently in openish beta in many places (Australia, Canada, Brazil, Japan, and plenty of other places). Graphically its got a very Diablo 1/2 vibe, and it's one of the few touchscreen action games I've played where the controls feel responsive. Online connection needed, due to the co-op nature of it.
Clash Royale is a fun little laning/strategy game based on the Clash of Clans franchise. Weird levelling/tiering system, but good to play for a quick competitive bash. I rarely play it at all these days, but it's certainly staying on my phone for when I do want to. Online only.
The rest of these don't need an online connection:
OpenTyrian has an android port, and is one of the best shoot'em ups ever. Plays surprisingly well on a phone. Laser/sonic ftw! Tap "Rear Weapon" to change it's mode, and don't use ammo-sidekicks and you'll be fine (which means no plasma storms unfortunately
)
The Trese brother's stuff has been mentioned. If you don't mind paying a few dollars, Templar Assault is a pretty good Space Hulk clone to scratch your 40k itch. Star Traders is a good trader/exploration/trek-like and Heroes of Steel is a nice RPG. Very god devs these pair.
The Ur-quan Masters is a port of Star Control 2 and is available on android. Damn good game, but a bit fiddly on phone. You can cyborg your battles, but that's half the game.
FreeCiv works pretty damn well on phone. Plus, you get to start with an explorer. This makes Civilisation so much more interesting at the start that it actually feels fun. Civ is pretty good anyway, but something to do at the start (other than expand) is great.
I'm pretty sure I got freeCol to work on my phone at one point (Colonization clone). From memory it was fiddly, and not a good port of it. FreeCol is great, just not the android version of it.
FreeHeroes2 is a Heroes of Might & Magic 2 port, for all your oldschool TBS needs. There's a HoMM3 port, but I haven't tried it yet. FH2 works fine, but you've got to be extra careful in battle for correct positioning sometimes.
OpenXcom is basically awesome. Hoping that they'll re-port it once Meridian's .exe mods get into the vanilla build, because a facing indicator would be superb. Xcom is always good.
Battlestation: Harbinger is a pretty good space roguelite. It's currently half price, and the engine is quite nice. Apparently it's been fixed up a fair bit in both balance and replayability-wise, so it might be worth a look.
OpenPanzer does the Panzer General games for you, if you like WWII strategy sims. I've had it on my phone for ages, but never really played it.
The Battle for Wesnoth has an android port. It's an old rpg/strategy game for PC. Heaps of scenarios, heaps of campaigns, hex-based goodness. D/L takes a while, but worth it. Slow to play, but good if you've got a few hours to kill.
For emulation, my two go-to games are Sid Meyers' Pirates! and M.U.L.E on the Nostalgia.NES emulator. Immensely replayable, and work great on touchscreen.
For dosboxing, I use Magic Dosbox. It lets you set up touchscreen icons for things like long-click, right-click, keyboard hotkeys, etc. This makes stuff like Simcity 2000, The Incredible Machine, Lords of the Realm, and even Warcraft (barely, but you can set touch-hotkeys for creating/selecting command groups) all be quite playable on a phone. It also makes it quicker to load your games, because they're menu icons now (no command prompt needed). Also very handy for win3.1/win95 games where right-click is vital. There's a free version for one-game-setup-at-a-time for trying it out, but the paid version is worth it. Probably one of the only things worth buying on android really.