That's sort of more of a music theory question than a ukulele-specific question. Without getting super technical about it, chords that are built off of a common scale will sound good together (that's more or less what it means to say that a song is in a particular key).
For example, the C Major scale goes C-D-E-F-G-A-B. A chord is formed by three (or more) notes being sounded at the same time. The most common basic chord consists of a root note, the note a third (either major or minor) above the root, and the note a fifth above the root.
The C Major chord is C-E-G. You can build chords using the C Major scale starting on any of the notes in the scale, though, which would give you D minor (D-F-A), E minor (E-G-B), F Major (F-A-C), G Major (G-B-D), A minor (A-C-E) and B diminished (B-D-F; don't worry too much about it yet). These chords are also referred to using the number that corresponds to the place of the root note in the scale that they're created from (the C Major chord is "the I," the D minor is "the ii", the E minor is "the iii", the F Major is "the IV", the G Major is "the V", the A minor is "the vi" and the B diminished is "the vii"). All of those chords will sound OK being played in the same song, because they're all derived from the same scale. It gets more complicated when you start extending the chords (adding sevenths and ninths and whatnot), but that's the basic idea.
There are some really common progressions that you hear in lots of songs. A I-IV-V-I is a common blues progression that you'll find all over the place in pop, rock, and jazz songs, too (sometimes as a I-IV-I-V-I or in some other arrangement). You often see ii-V-I progressions in standards - sometimes with variations (vi-ii-V-I, iii-vi-ii-V-I, and so forth). These are common progressions (in part) because they all build tension that gets released with the return to the I chord from the V chord.
Howlin' Hobbit put together a good selection of common chord progressions for ukulele
here. Check it out and see how they sound!
