A chord does not have to be made up of thirds. A chord is by definition two or more notes heard as if sounded simultaneously. Not all chords have three notes either. There are dyads (two notes), triads (three), tetrachords (four), pentachords (five), and hexachords (six). There's no limit on the number of notes, and also, by definition, there's no limits on which notes.
C - E - G is a chord.
D - E - F - C is a chord. However, the most common triads are the major, minor, augmented, and diminished (there is also the suspended). All of these are composed of a root, a third, and a fifth (except the suspended, which uses the root, perfect fourth and perfect fifth).
So, now to your question, why thirds? First, realize there are two types of thirds: the major and the minor. The major consists of four semitones and the minor three semitones. Quoting Wikipedia:
The major third is classed as an imperfect consonance and is considered one of the most consonant intervals after the unison, octave, perfect fifth, and perfect fourth. In the common practice period, thirds were considered interesting and dynamic consonances along with their inverses the sixths.
After the major third became established as such, it become pretty standard. Every classical piece makes use of it in some way. The other reason the major third is so widely used is that it is found in the harmonic series (between the fourth and fifth). Early brass (e.g., posthorn, natural trumpet) had no valves or slides and were limited to the harmonic series. This encouraged use of and familiarity with the major third. However, I'd say the most important of all these reasons is the first. It is highly consonant.
The minor third has the same level of consonance as the major third, but is found higher up in the harmonic series (between fifth and sixth). Also, there are many common transposing instruments which sound a minor third higher or lower form where they are written. For example, the Eb clarinet and the Eb trumpet both sound a minor third higher than written. The oboe d'amore, popular in the eighteenth and twentieth centuries, and the soprano clarinet in A sound a minor third higher than written. Of these reasons, I'd say the first (again) is the most important.
As for other intervals, any interval can be used, but some are more common than others. The perfect fifth, octave, unison, and seventh (in no particular order) are very common. All major, minor, and suspended chords have a perfect fifth. Also important are the perfect second, perfect fourth, and major sixth. To learn more about different chords and the intervals that make them up, read this article on intervals and this one on chords. They are both very informative.
Thirds are the most consonant intervals (after the unison, octave, perfect fifth, and perfect fourth).
Are other intervals sometimes used?
Many other intervals are used. See here for a list of the main ones. They include the perfect fifth, the perfect seventh, the octave, the major sixth and the perfect fourth.