The Mandolin is mostly used for playing melody lines, but chords are equally possible.
To produce a chord, you have to press the right string courses on the right frets, in such a way that only notes from the chord produce a sound.
Say for example, that you want to make a Cm7 chord. This chord has the notes: C Eb G Bb, so you would proceed as follows:
The following image shows this chord:
Following this logic, you can construct all chords, once you understand the theory behind intervals, and chord formulas.
This is fine when you have plenty of time in your hands, but when you just want to play a song with some friends, grab a copy of the DG Mandolin Chords application for Android, which will show you how to make chords quickly.
We recommend you learn by memory the basic major, minor, and seventh chords. The rest comes with practice as you need them.
Also learn some basic chord progressions.