This achievement becomes possible once you recruit Mordin Solus for your team, at which point you can access the upgrades console in the Tech Labs on the Normandy.
Upgrades can be found during missions by looking around, by purchasing them at stores, or by consulting with your team members on the Normandy, who often have special upgrades to add. Upgrades that you've purchased at a store immediately go into effect and count towards the achievements. Upgrades found in the field or given by crew members, however, have to researched in order to count toward the achievement (this will also make them go into effect). This requires mineral resources and can be done at the research terminal in the Tech Labs on the Normandy (where Mordin is), or by selecting the upgrades option in the dialogue wheel when talking to members of your team on the Normandy. Once you have researched or purchased 10 total upgrades, the achievement will unlock.
A list of all upgrades (and weapons) to be found in the field (remember that you have to research these on the Normandy for them to go into effect): http://www.gamefaqs.com/boards/genmessage.php?board=944907&a...
You don't necessarily have to go out of your way for this achievement. 10 upgrades is a rather small amount, and you can likely get this without any major sidequesting. If you want to be extra cautious, the link posted above will help.