I am first and foremost a song lover. Chords (the importance of which is not taken seriously enough by many songwriters imo. Listen to the changes in "God Only Knows" by Brian Wilson, and "What Becomes Of The Broken Hearted" by William Weatherspoon, Paul Riser, and James Dean. Glorious!), a melody (or three, if it's a fugue), harmonies, counterpoint if you're really good. For a song to be heard, you usually need some instruments to play it, A cappella being the exception. Different musicians approach the playing of a song in different ways; some view the song as merely the platform from which they express themselves via their particular instrument. Others play their instrument in service to the song itself (and/or the singer and other musicans), their talent at that assessed by how much the song benefits from their playing of it. I am a fan of the latter approach.
The players I like most can be described as "lyrical"---their parts sound like an integral element of the song---it would not be the same song without them. These type players are highly valued by the best songwriters, producers, and singers, and by other musicians who also play via that approach. Rick Hall, owner of the infamous Muscle Shoals Sound Studios, said the members of the studio's house band---The Swampers---all thought like and played as if they were producers. Exactly! When a great song is played by these type musicians, backing a good singer, the results are magical.
My favorite players are guys like Jerry Douglas (dobro), a member of Alison Krauss' Union Station and a first-call Nashville studio player. Ry Cooder and Richard Thompson, master guitarists of course. There are plenty of others, but that's enough outta me. Anyone who wants to add to my short list, may I request we keep it to lyrical/musical players, not "hotshots"? They already get all the attention (and chicks ;-) .