Jazz standards are often shared as "charts" or "lead sheets", simplified sheet music that show a piece's melody, annotated with chord symbols. This information is enough for jazz bands to coordinate and play complex music together, or enough for a competent jazz pianist to perform an ornate piece for solo piano.
I'm going to talk about some methods that a jazz pianist/composer might use to go from a lead sheet to a fully realized piece of music.
:: Harmony, Voicing, Chordal Movement
1. Closed voicings, Root position triads and tetrads: The most basic thing you could do with the chords written is to play them in root position below the melody, voiced exactly as they are notated, i.e. tertian voicings, with occasional alternations for suspended chords, sixth chords, missing upper chord tones, et cetera. For the most part, you'd play ^1, ^3, ^5, ^7 in that order, and you just hold the chord until the next one comes along. Super boring.
2. Closed inversions: Instead of playing the root note constantly in the bass, look at the melody when a chord appears. If, when the chord comes into play, the melody has a chord tone of that chord, then play whatever inversion of the chord puts the melody note at the top. If the melody doesn't have a chord tone when the chord comes in, find the first note of the melody that is a chord tone, and play the lower notes like the inversion that fits it, and then just move the upper note with the melody. You can also chose inversions on the criterion of minimizing how much your hands move over the keyboard, for smoother voice leading at the expense of not nestling up so tightly to the melody. But maybe the melody needs some space.
3a. Shell voicings: The next trick you might try is to play the chord more spread out over two hands, perhaps playing ^1 and ^3 or ^1 and ^7 in the left hand (a "shell") and other chord tones in the right hand. Dropping ^5 from the chords entirely, when the chord contains a perfect fifth, is another good way to create a more open, spread out voicing in jazz. You can also play a shell in the left hand and a triad/tetrad inversion in the right hand to match the melody.
3b. Drop 2 voicings: If you start with a closed triad/tetrad or one of its inversions and you drop the second note from the top - down it down an octave - that's a drop 2 voicing. They're a very nice way to give you a more open sound without too much thinking. The dropped note will be your bass note, instead of a shell. This is an alternative to shell voicings, not really a step more or less advanced. They're both good techniques. You can use one or the other in different sections. Once you're familiar with them, you might try combining them to see if you can get smoother voice leading from the combined space of open chord voicings.
4a. Barry Harris chordal movement: Especially when a melody moves by 1st and 2nd intervals, there's a technique for chordal movement in the right hand taught by Barry Harris that is easily applied. Every note of such a melodic phrase has a chord under it. Suppose the chord indicated over the phrase is C.maj6. The scale which matches this is the major bebop scale, [C, D, E, F, G, Ab, A, B]. If you play alternating notes of this scale, you either get inversions of C.maj6 (i.e. [C E G A]) or inversion that are inversions of B.dim7 (i.e. [B D F Ab]). When you have a melody that moves by step, you can apply one of these chords below each melody note, usually alternating between a C.maj6 inversion and a B.dim7 inversion. In the Barry Harris system, other chord types are associated with other scales. I'll write more about it here soon, but you're just swapping between inversions of two chords and using that to make chordal runs in lockstep with the melody.
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4b. Partial inversion: Complex chords with upper extensions can often be played as one triad/tetrad over another triad/tetrad. The benefit of thinking of them this way is that triads and tetrads are invertible, whereas chords spanning more than an octave (9th chords, 11th chord, 13th chords) are not. So while your left hand is fixed playing one thing, the other hand can move through inversions of its notes (and passing chords between the inversion). The Barry Harris chordal movement system is a simplified form of this that doesn't require much thought, but you can find inversions within more types of chords than Harris specified and you can use more kinds of passing chords than dim7.
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5. Functional Extension Of Chord Progressions: Chord Progressions have structure, and you can use the rules of chord progression structure to add in more chords between notated chords. In this way, you could have a chord on every beat or every eighth note or any granularity you want. Alternatively you could have a chord for every melody note, and thus play with the same rhythm as the melody.
:: Comping, Rhythms, Bass Lines, Counterpoint
Above we saw good methods to realize melodies with notated chords as complexly structured musical segments, approaching something like a vocal chorale at the higher levels of complexity. There are other principles of jazz composition worth using as well, like grooves that play a chord with a specified rhythm or suggested stress over an extended period of time, low notes here, high notes there. Bossa Nova and Samba have beautiful chordal rhythms like that. Stride piano is another formulaic method of adding rhythm to your chords, if you like ragtime jazz. I think that a lot of the technique for adding rhythm goes mostly unnamed in jazz: you see a piece that accents 1 and 3+ or something, and it's easy enough to internalize and apply to other songs, but it's not complex enough to be worth calling "the groove made famous by Snappy Snow Peas". Simple rhythms like these are especially useful if you're accompanying a solo, and you either need something simple to do in the left hand while you focus on the in your right hand (staying out of your own way), or if you're in a band and want to stay out of another soloist's way.
You can also make up bass line and countermelodies. The way to compose melodies and to intertwine them contrapuntally is perhaps the densest, most theoretical, most computationally demanding aspect of music composition, and I will not be talking about it much here.
I will say that as you become familiar with melodic lines and bass lines in jazz, you can start to develop a vocabulary of melodic motives that can be strung together to form sentences. Moving chromatically, diatonically, or chordally, and moving in any of these ways up or down, these are the obvious components of most melodies, but when I talk about vocabulary, I mean something more structured. For example, you might know some riffs for connecting two notes an ascending diminished fifth apart, or a descending major third, or other intervals. By stringing these up, you can make longer melodies and phrases. The location of chord tones in the voiced chordal outline of a piece gives you landmarks to navigate between with riffs like these. Here are some tertian licks I learned from Shan Varma:
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They're a great way to get started with jazz soloing. They might even have application in composing jazz bass lines.
One of the simplest examples of formulaic connection of chord tones for melody generation is the walking bass line. Despite being very simple and formulaic, it is still a very useful compositional form which works against melodies even when you compose it with basically no consideration of counterpoint. Let's cover the method here briefly.
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Other simple bass line composition techniques are, I think, much like chordal comping rhythms, mostly unnamed. You find a rhythm, you play one or two low notes with that rhythm, you apply it in lots of places. It's not complex, but it gets you through a bar of A.m7.
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In Afro-Cuban music, there are rigid rhythms that that guide a piece, sometimes a whole genre. I don't know enough about these, but if you look up son clave, montuno, and tumbao, that will be a good starting point to learn more. Within this system, there are rhythms and other structural suggestions for bass lines and chords and arpeggios. You could try something like that if you want to add rhythm and bounce to a lead sheet.
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