In most European/Western music since some point in the baroque onwards, chords have been a rather central concept. The chords provide a background harmony, which it is easy to write melodies over. Chord progressions provides the song with a sense of things advancing, and chord-based approaches make it easy to change a key by a variety of methods.
Chords are compound tones - they are several tones played at once. Depending on your experience with music, you may know that, i.e. C#major is C#,F,G#, and that Fmajor is F,A,C. Now, some hobbyists may be under the impression that chords are labels for essentially arbitrary groups of tones, but they are in fact a systematic terminology for groups of tones that have certain relations.
The most basic chord in western music is the major chord. Each tone in the 12-tone temperament can be root for one. The major chord has a rather interesting structure - in just-intonation (where intervals are basically tuned like intervals between members of the overtones series), the major chord would be 4:5:6. That is, [4/4, 5/4, 6/4] times a shared fundamental frequency. If we were to pick A=440hz as our starting point, we get [440hz, 550hz, 660hz]. A similar structure is present at [3/3, 4/3, 5/3], which gives [330hz, 440hz, 550hz]. We notice that 330hz is an octave below 660hz, so we can consider them the same structure. [5/5,6/5,8/5] too gives a similar structure with two of its members offset by an octave. Structures where a member is offset by an octave are referred to as 'inversions' of the same chord, as the same equivalence classes of tones are present (i.e. the classes C,E,G are present regardless if it's or C3, E3, G3 or E3, G3, C4.) Some experiments indicate that musicians hear inversions as the same chords, whereas people who don't play instruments tend to hear them as distinct chords. This may be worth being aware of when considering the music of your culture - apparently, it's a thing western musicians learn to hear them as, and thus it is entirely conceivable that some culture would have musicians that maintain the distinction.
We could write a song where we just stick to one chord throughout it, and the melody is the main source of motion, tension and interest. Such songs exist. However, it is more usual once we start using chords to create motion, tension and interest by changing the chords - and letting the melody create further motion, tension and interest against the backdrop of the chords.
So, we may want to build new similar structures from other tones in the scale, say we pick the topmost member of our first chord as the root of the new one; we could also pick the bottom-most member of it as the topmost member of a later one, obtaining:
440*(6/4) * [4/4, 5/4, 6/4] = 660, 825, 990440*(4/6) * [4/4, 5/4, 6/4] = 293,333..., 366,666..., 440
gives us all of these, when reduced to the same octave: 440, 495, 550, 586,666..., 660, 733,333..., 825, 880
The same procedure can be applied to any frequency to obtain the frequencies of the major chord rooted on that frequency. For reasons I will get into soon, having chords tuned exactly to these ratios causes certain problems, and this is solved by only approximating them - this benefits from our hearing not being all that exact.
Now, I mentioned earlier that multiplication is to a number what addition is to its logarithm. So, if we take logarithms of [4/4, 5/4, 6/4], we can now add things instead, by using cents:
[0, 386, 702] + [702, 702, 702] : same thing as multiplying [4/4, 5/4, 6/4] each by 6/4.[0, 386, 702] - [702, 702, 702] : ... dividing by 6/4
Although hertz numbers are easy to understand, they're less general than cents are. Cents are easier to work with than ratios though, because a) it's easier to compare sizes - which one's bigger out of 19/16 and 6/5? How about 298c and 316c? - b) it's easier to add than to multiply. Octave reduction is done by subtracting or adding 1200 (repeatedly, if so needed), c) since the cents-value basically is based off the twelve-tone tuning of western music anyway, an interval measured in cents tells us something about what familiar musical intervals it is close to - i.e. 25/16 and 8/5 might not spontaneously tell us much - but 772c and 813c tell us they're both in the vicinity of 800c, so fairly close to the interval you get by stacking eight minor seconds - C to Ab, D to Bb, etc.
Chords need not be 4:5:6, though - we have minor, diminished, extended chords and sus chords in western music, and there are other possibilities as well.
Minor turns the order of the intervals upside down: from [4/4, 5/4, 6/4] we go to [6/6, 6/5, 6/4].
((6/4) / (5/4) = 6/5). The minor chord famously is 'sadder' than the major chord, and is the other main staple of chords. If you use some specific type of triad, you should probably try and see if its ~inverse sounds appealing to you as well. For [3/3, 5/3, 7/3] this would mean [7/7, 7/5, 7/3]. They can be recalculated to have a uniform denominator, i.e. [6/6, 6/5, 6/4] can be rendered as [10/10, 12/10, 15/10,] and [7/7, 7/5, 7/3] as [15/15, 21/15, 35/15].
Sus2 are [8/8, 9/8, 12/8], sus4 are [6/6. 8/6. 9/6] (or in the other notation 8:9:12, 6:8:9). Diminished and augmented chords are slightly less clearly any one thing and can stand for several possible interpretations. The diminished chord can easily be seen as [5/5, 6/5, 7/5] or [7/7, 7/6, 7/5] or [6/6, 6/5, 36/25]. A different musical tradition may have a more specific idea of what a dim chord is, or even have several different dim chords.
Extended chords sometimes are understood as going further up the overtone series, so a seventh-chord adds one tone higher (depending on type, you usually get something close to [8/8, 10/8, 12/8, 16/9], [8/8, 10/8, 12/8, 15/8] or [4/4, 5/4, 6/4, 7/4]. The first is roughly or dom7 chord, the second the maj7 chord, and the third is common as dom7 in Barbershop and some blues. Again, by sheer luck, 9-chords too generally add a 9-like number: [4/4, 5/4,6/4,(whichever seventh the chord name specifies),9/4] (although b9 may add things like 17/8 or 32/15, #9 may add 12/5 or 19/8 or somesuch, and so on).
Chords with the extensions 11 and 13 are further off from the corresponding overtone when played in equal temperament, however. Using extended chords as "proper" chords has some roots in Baroque times (where some 9-chords were definitely used in a chordlike manner, e.g. dom7b9, which for E would spell out E-G#-B-d-f, a strong dissonance that goes well in minor keys, so e.g. Edom7b9 would appear in A minor compositions, Cdom7b9 in F minor, etc. In the 19th century, especially major seventh chords became increasingly considered consonances, and in the twentieth century especially jazz has made some quite wild chords acceptable as 'chords proper'.
Other series could well be used - Bohlen-Pierce-scale composers draw a lot of mileage out of 3:5:7 and 5:7:9, and some microtonal composers utilize things far up the overtone series, such as 14:17:21, 12:17:23 or things like that.
It is possible to build your chords from a subset of the scale used for melodies; the opposite is possible as well - lots of western popular music take its chords from all seven notes of the diatonic scale - or even more than that - yet restrict the melody to the pentatonic scale that is a subset of the seven notes. Some music, however, restricts its chords to a smaller set than it restricts the melodies to. So both approaches are apparently fertile ground for melodic/harmonic interaction.
Not all music classifies simultaneously played notes as any kind of chord; even in the baroque era, such a classification system was not all that widely used. Renaissance polyphony definitely does not use the same kind of chords as we do, but would have considered all three-note chords except root-fifth-octave as dissonances that should be resolved (yet did use combinations of three or more tones as a device for tension). Polyphony in general can use tone-combinations without giving names to them - there are lots of possible combinations, and naming them all may seem rather tedious and pointless.
[note: I will add some samples of chords of various kinds later]
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