Applications

Downloading Anime and Manga

uTorrent for Torrents
Direct Downloads with your favourite Webbrowser
Tucan Manager JDownloader for files hosted by FileHosters like MegaUpload, Rapidshare and Mediafire

Watching Anime

CCCP optionally with CoreAVC for watching h264 encoded videos
and using Media Player Classic - Homecinema as your default video player

Reading Manga

MangaMeeya or CDisplay or CDisplayEX

File Verification

RapidCRC or RapidCRC (continued by ov2)

IRC Clients (also for downloading Anime and Manga)

XChat official or the free XChat-silverex build ( see my configurations for easier leeching )
You might want to also check out HexChat which is a maintained fork of XChat.
mIRC is more popular and has a wider support