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All of the following are totally free. Links are not provided to any software listed here because it can all be directly obtained without needing to visit each site individually, searching for the download link, and then running an installer. Painless installation and automatic updating are built-in with all of these.

Alarm:
Just use crontab and your favorite media player and music.

Antivirus:
First off, lol.
ClamAV is fine if you really want antivirus software.

Audio:
Amarok
Banshee
Dragon
Juk
Exaile
Rhythmbox
VLC
(plus countless more)

Audio Editing / Recording:
Audacity
Ecasound (for batch audio editing)
Jokosher
Muse

BitTorrent Clients:
The original bittorrent client is native in Linux as well, and uTorrent works fine through Wine.
Transmission, Deluge, and kTorrent are also all fantastic.

CD/DVD Burning:
Brasero

Compression:
Just use tar and gzip.
Alternatively, the "p7zip-full" and "p7zip-rar" packages will handle silly proprietary compression schemes your Winblows-using friends will send you.

Conversion:
ffmpeg beats all, no contest.

Download Manager:
wget will kick the crap out of any download manager you've ever used.

Gaming:
Wine runs Steam and most popular Valve games as though they were native.
Plenty more work besides, and updates to Wine tend to be focused on popular games, but Linux is obviously lacking when it comes to new releases.

IM/Chat:
Empathy and Pidgin both pretty much cover all bases.
There's a native Skype client for Linux, too.
TeamSpeak is an excellent VoIP server/client free from the fascism of Ventrilo, but Mangler is a Linux port of the Ventrilo client in case you want to talk on Vent.

Image Editing:
GIMP
I'm not big on graphics, but there are undoubtedly other vector graphics packages and all sorts of whatnots. Wikipedia tells me that Inkscape is good.

Internet Browser:
Firefox
Google Chrome
Opera
Dozens of others besides.

IRC Client:
Countless numbers, including the already mentioned XChat and Chatzilla.
tinyirc, Bersirc, BitchX, lostirc, and KVirc are also all excellent clients.
And of course, the best of the best: irssi.

Office Software:
Eeeewwwwwwwww.
OpenOffice if you absolutely must use Office for some retarded reason, or IBM Lotus Symphony if you like the candy-covered look.
Typeset your work with LaTeX, and the world will take you much more seriously. Not to mention LaTeX is practically a requirement for math/science/engineering grad school and industry work.

OS Emulation:
Sun Virtualbox is native in Linux, as is DOSBox.

PDF Reading:
evince
xpdf

Video:
VLC is native in Linux, as is Miro.
mplayer, Ogle, xine, and Totem all rock the Casbah.

And we didn't even get into web servers (apache2, lighttpd), remote file-transfer (scp, sftp, lftp, rsync, sshfs), remote login (ssh, VNC), systems administration (NIS/NFS, OpenLDAP, Kerberos), version control (mercurial, subversion, git), encryption (GPG, mcrypt, TrueCrypt), mathematics utilities (Octave, gnuplot, scipy, orpie), or the glory that is the shell (zsh ftw!).

And it's not surprising that you didn't even mention customizing the UI, because Winblows users and Macfags are not used to that even being an option. There's a stupidly huge list of windowing managers, all open-source and easily extensible, which are also not integrated into the goddamned kernel (a totally senseless limitation brought to you by Microsoft).