I use the apps listed below a lot, and couldn't imagine my android life without many of them:
Music player: Shuttle+, because of its holo UI, simplicity, great EQ presets, DSP support. Worthy mention: GoneMAD Music Player.
Launcher: Nova, because it's basically stock UI cranked to 11 and with wide array of additional functionality. Apex is essentially the same, but my heart lies with Nova. Worthy mention: Smart Launcher, I like its different approach, simplicity and concept.
Automation: Llama, because it's a very powerful tool for people who think Tasker is too much/too hard.
Multitasking: GYF Side Launcher, because it works and feels the best for me, has just the right configuration options and is under constant development. Worthy mention: Sidebar, constantly adding new features, attentive to user requests/comments, works great.
Weather: Eye In The Sky, because it's simple, uncluttered and unobtrusive. Worthy mention: 1Weather, for the same reasons.
RSS/News: Taptu, mainly because it lets me merge streams and thus maintain clean and uncluttered look. Not too flashy, which I don't like. There might be another RSS/news app which lets you merge streams, but I haven't seen it.
Some random apps I use a lot for various purposes and I think are definitely great:
SofaScore for sport results,
Unified Remote for controlling so many things from your couch,
AdAway which is self-explanatory,
BetterBatteryStats for hunting down those elusive wakelocks and monitoring battery usage,
Greenify is the best you can do for your battery instead of the useless app killers and the placebo battery savers,
Solid Explorer is my file browser of choice,
SuperBeam for quickly transferring data between devices on the same wifi network or using NFC,
TuneIn Radio for a great array of (internet) radio stations to listen to,
WiFi Analyzer helps me to configure my network in a more efficient way.
Hope this helps someone to discover some new great apps!