MP3/OGG Music[edit]

Audacity Open Source Windows/Linux Sound Editor
LMMS Open source synthesizer and sequencer. Available for Linux, BSD, MacOSX and Windows. Large built in set of instruments; Variety of synthesizers; Wavetable synthesis, SoundFont, VST, LADSPA, and MIDI-in support. Renders to MP3 or OGG, among other formats.
If you want to convert MOD music to OGG/MP3, several mod players support a 'wav' output driver which dumps the sampled music to wav, which you can then encode into the desired format. If your soundcard has a 'capture' channel, you may also be able to capture the audio output as it's playing, via Audacity; You'll probably have to manually select the Capture channel as the mic input device for this to work.
Csound Extremely powerful open source synthesizer, sequencer, and more. Cross-platform.
SoX A command-line program to convert and apply effects to sound files of various formats. "The Swiss Army knife of sound processing programs". Cross-platform.

Sound Effects[edit]

sfxr provides a simple sound generation system that is capable of producing almost any simple 8bit style sound effect; For more complex sound effects you can combine sound effects with your sound editor.
- bfxr, and 'elaborated'/extended flash based version
It's also possible to use a tracker like FamiTracker to create sound effects!
Use one of the tools in the above section to edit your sound effects files as necessary

ModPlug Tracker Free. Can be as simple or complex as you need it to be.
MilkyTracker Open source, Linux/BSD/MacOSX/Windows/(other platforms. Intended as a FastTracker 2 clone, so best for composing .XMs. Imports a wide range of formats, exports MOD, XM, or WAV (so can be used to help convert modules to MP3/OGG format). Supports MIDI input.
AmigaMML Open source, cross-platform. Exports MOD and XM formats. Compose music using Music Macro Language.

Anvil Studio Shareware Windows MIDI composer
Rosegarden Open Source GNU/Linux MIDI composer
Aria Maestosa, a MIDI editor which covers all basic needs and has a simple interface. Has various editors including percussion and score editors.
A list of MIDI sequencers
Add more!

Musagi, by Dr. Petter of sfxr fame, is a tracker somewhat like ModPlug, but with more flexibility. Can be used to create NES- and Commodore 64-like chippy songs, or full fledged songs, which can be saved to WAVs. Use Audacity or SoX to convert to .ogg. Here's a good tutorial for beginners.

Wolfram Tones Can generate surprisingly good random music based on genre, which you can then tweak with a set of included online tools.

Bosca Ceoil is a very easy to use music editor that can be used right in your browser