For the most part translation patches will come in one of a variety of patch formats, some will come as programs to do patching (which some anti virus programs have problems with for no good reason so be aware of that).
Once patched you will be left with a .nds ROM which you copy and play like anything else. Depending upon the setup you might have to alter your cheat database to have it work with the altered game but that should be a matter of just getting the new identifier with R4CCE or whatever you are using. Most translations will come with a nfo file/readme/release notes that tell you what goes. Do not bother with prepatched ROMs as a lot of them are older versions of hacks/translations with unfixed bugs.

As mentioned there are a variety of patching formats used by various hackers and such like

xdelta is arguably the most popular. There are two versions
http://xdelta.org/ is the current one and used by most newer translations. Go with this first.
http://www.evanjones.ca/software/xdelta-win32.html is based on an older version
There are GUI programs for it like http://filetrip.net/nds-downloads/utilities/download-xdeltaui-mod-1-0-f17554.html

bsdiff. Back in the "real" world this is the big rival to xdelta. http://www.daemonology.net/bsdiff/ has it. Not so many hackers used it on the DS but it was popular among some Scene groups for trainers and such like.

Older systems used a format called IPS. It struggles with files larger than 16 megabytes or so though and even the GBA had stuff bigger than that. I have seen it (probably even used it myself) a handful of times when individual files need to be patched. There are a million different IPS patchers. http://home.arcor.de/minako.aino/ipsXP/ and gbata from http://www.no-intro.org/tools.htm are my usual two though you may have to force it to view all files.

Some use custom patch formats, usually for good reason but release notes will tell you that. The only you are likeliest to meet first is the one used in deufeufeu's Jump Ultimate Stars translation ( http://filetrip.net/nds-downloads/rom-hacks/download-jump-ultimate-stars-030508-f30428.html includes a download).

Other formats. More common on older systems and the wii for some reason was a format called PPF (playstation patching format). http://www.afterdawn.com/software/cd_dvd/iso_tools/ppf-o-matic.cfm is a patcher. There are multiple competing takes on the PPF concept so be aware of that.

I could go further but if you are only consuming hacks there is not much point as I just covered most things worth knowing.