One step at a time:
First off, most chords are built on a "triad"; a set of three pitches, with a predefined interval between each of them, beginning at a specific note. This beginning point is called the "root" of the chord, and it is the note for which the chord is named. This is not always the note on the "bottom" of the chord; we'll explore that later.
For instance, a "C" chord will have, as its root note, a C. The chord's name and quality are the same regardless of which octave, or how many, are present in the notes played.
Now, the triad, built on the root, will have predefined intervals between the three primary notes of the chord, which when based on a particular root, will determine what those actual notes are and what the resulting "quality" of the chord will be.
The most basic triad quality is "major". Major chords have, as their defining tonal characteristic, a "happy" sound. The major triad is composed of the root, then the note a "major third" (two "whole steps") above it, and then the perfect fifth (which is a "minor third", a step and a half, above the second note of the triad). A C major triad (simply labelled "C") is composed of the notes C, E, and G, which are, for the same reasons, the first, third and fifth notes of the C Major scale.
The next most common is the "minor" triad. Minor chords have the same root and perfect fifth, but the middle note, the third, is lowered by one half step. This makes the interval between the root and the third a "minor third", meaning the interval between the third and fifth is now "major" (because that is as I said the same note). Minor chords, compared to major chords, are generally described as sounding "sadder". A C minor triad (labelled "Cmin" or "Cm") would be composed of C, Eb, and G.
After that, the last "natural" triad (composed of three notes, each separated by two "scale degrees" from their neighbors) is the "diminished" triad. This occurs naturally in the major scale when you take the seventh, second and fourth scale degrees as a triad; both of the intervals between the three notes end up being minor, so not only is the third lowered by a half-step, but so is the fifth. Diminished chords are generally described as sounding "dark" or "sinister"; many of the cliche examples of organ music the bad guy would typically be playing, like Bach's Toccata in D and the theme to Phantom of the Opera, have a lot of diminished chords. A C diminished triad (Cdim) would be composed of C, Eb and Gb.
There are other possible triads. The most common "altered" triad is the suspended triad. Instead of using the third scale degree as the middle note, you use the fourth. This is a very powerful chord, because it contains all three of the notes in a major key from which you can form a major chord by using it as the root, and it also sounds like it "wants" to resolve to the major triad. A C suspended triad (Csus, sometimes Csus4) is C, F and G
Beyond triads, many chords commonly used in modern music add additional notes to the triad to form a four-note chord. The most common note to add is the seventh scale degree, however it is typically lowered by a half-step to produce the "minor" or "dominant" seventh. A C dominant seventh chord (notation varies; most commonly it's C7) is C, E, G, Bb. The dominant can also be added to the minor chord (forming a minor-minor seventh; Cm7), so that's C, Eb, G, Bb. The major seventh can also be added to either of these, forming the major-major seventh (CMaj7; the use of "maj" here is different from the triad) or the minor-major seventh (CMm7), C-Eb-G-B.
Lastly, the seventh, added to a diminished triad, forms a diminished chord. If you add the dominant/minor seventh, the chord becomes "half-diminished" (CØ7), C-Eb-Gb-Bb. The major seventh is typically not added to a diminished chord (though if you're after a very dissonant sound, this will certainly do the trick).
If you instead add the sixth (or the double-flatted seventh aka the "diminished seventh") to a diminished triad, the chord becomes "fully-diminished" (CO7), C-Eb-Gb-A. Mathematically, the fully-diminished chord is very important in music theory, because all of its notes have the same 1.5-step interval, and because there are only three unique combinations of notes that form a fully-diminished chord. C-Eb-Gb-A, Db(C#)-E-G-Bb(A#), and D-F-Ab(G#)-B. The next higher diminished chord, formed by Eb-Gb-A-C, has the same notes as the first; you may call it EbO7, but it's equivalent to the CO7 chord, it just has a different note on the bottom. It is, as we call it, an "inversion".
An inversion is usually notated the same as the normal triad, but instead of the root note being on the bottom, one of the other two are. The "first inversion" is formed by taking the root note and playing it an octave higher, so the third is "in the bass". The second inversion is formed by also playing the third an octave higher, so the fifth is in the bass. As I said, the notation usually doesn't change, however if it is important that a particular note be in the bass, it may be notated with a slash followed by the bass note: a first-inversion C major triad might be notated C/E, while the second inversion would be C/G.
Inversions and other "alternate voicings" are extremely common in music, especially pop music, for two reasons; first, because music in many forms is thought of as being formed by the movement of independent "voices" up and down by individual scale degrees, and second, because on many instruments, notably the guitar, it's difficult to play most chords with the root note "in the bass". You instead usually play the combination of notes however you can. In chords spanning multiple octaves, these defined inversions lose most of their meaning; you typically just start on the bass note (or any note in the chord you can sound) and play every note in the chord in every octave that you can until you're told by the sheet music to stop or you run out of strings or keys.
The sixth has become a more common addition to triads with the popularity of jazz. A C6 chord is a major chord with the sixth added on, often in an inversion with the sixth in the bass, where it forms a minor third under the root. Adding the second (also called the ninth) is also seen sometimes, especially in guitar music (a common way to form the C chord in the key of G will keep the D note from the previous chord; this is called a Cadd9 or a "country C").
Beyond this, most chord notation systems begin to break down, as representing "cluster" chords and others not based on a typical triad requires basically spelling it out.