A torrent seedbox is a permanent 24/7 online high speed server dedicated to download and upload (aka seeding) a torrent for you, an average seedbox has a 100 Mbit/s line or more, which is considerably faster than any home ISP and gets around bandwidth throttling if there is one, after a torrent has been downloaded onto the seedbox the user gets it to his computer via S/FTP, this is the only bandwidth consumed by the user, seedbox bandwidth does not count towards your monthly download capping if you have one.
For a monthly fee a seedbox provider will give you an always on server with a dedicated 100Mbps line, dedicated IP, huge amount of bandwidth, an FTP account, allocated hard drive space and RAM with an easy to use interface like TorrentFlux-B4rt to seed your files.

Seedbox torrent interface
Private trackers require users to keep a good ratio to be able to download files from other members faster, instead of leaving your computer on all the time and slowing down your Internet connection, a seedbox will free up computer resources for other tasks, if you need to distribute a torrent file making sure that it is available at all times, a seedbox is the way to go.
The most expensive seedboxes can have a VNC (Virtual Network Computer) remote connection to a Windows machine allowing you to run any BitTorrent client you like, they should also offer torrent creation, unRaring, ZIPing, and firewall protection, the online space they offer is meant to be used for seeding the files but there is nothing stopping you from using it for storage, OpenVPN and Usenet access should be available in the costly high end servers.
Any computer can be converted into a seedbox, with a little knowledge of Linux command line you could easily set up your own seedbox on a VPS as long as their terms and conditions allow it (high bandwidth and resources needed). You would probably not save any money using a VPS compared with renting an specialist seedbox provider, the only advantage of setting up your own seedbox on a server is higher privacy because you are in control of everything, logs and configuration, it is possible to set up a seedbox at home but your ISP might ask you to upgrade to a business line when they detect high bandwidth usage.
Seedboxes and sharing copyrighted files
A seedbox IP can be tracked down just like your home computer, but instead of serving your ISP with a DMCA take down notice they will serve your seedbox provider and they have to comply with it, a seedbox it is not a way to bypass copyright laws, if you use an open tracker it will be found, when a seedbox provider gets a DMCA notice they forward it to you asking to remove the infringing content, DMCA notices are simple take down notices, not court orders requiring personal details, it is when a DMCA notice is ignored that courts can get involved.

Torrentflux seedbox interace
If you make sure that your seedbox is located in a country with weak filesharing copyright laws, one of the few left is Canada, it won’t be possible to serve them with any take down notice even if the customer lives in France (three strikes law), or the US (lawsuits). Anyone sharing extreme illegal files on a seedbox will be caught, in such hypothetic scenario the seedbox provider would be served with a law enforcement subpoena and not a DMCA notice, subpoenas ask for customer personal details and logs that will have to be provided.
What to look for in a seedbox
There are lots of seedbox providers, look for a well established company with good reputation, make sure the specs fit your needs, bandwidth is not normally a problem but seeding files could be as the cheapest seedbox plans impose a limit on the number of files you can seed, get a company with a money back guarantee, you will get discounts for paying for a year, the wise thing to do is to try them out for one month first before any long term commitment.
Why to get a seedbox?
To make a file available 24/7 on the torrent network, to avoid leaving your computer switched on all the time, to speed up uploading and downloading of files, to keep a good filesharing ratio, you are on an ISP that caps bandwidth or you are a member of a private tracker.
Seedbox providers:
Seedbox (USA company): http://www.seedm8.com
Seedbox (French company): https://www.seedbox.fr
Seedbox (Dutch company): http://www.yourseedbox.com