One question I'd like to ask is where you came at drumming from. If you picked up the sticks because of listening to popular music and you are starting out on your own, chances are you are approaching playing drums from a more mechanical standpoint, i.e. I hit the hi-hats two times, on the third one I also hit the snare, then the hi-hat three times, etc. To some extent there is a physical, muscle memory component to playing drums -- probably more than any other instrument -- but without the headspace behind what you're doing, it can be hard to introduce musicality.
I lucked into musicality because I came to drums by way of several other instruments, a strong music program with a series of excellent teachers who emphasized reading and music theory skills, and taught us how to listen to music -- all kinds of music. Not everyone gets this sort of opportunity nowadays, especially since music programs in schools have often been targets of budget cutting here in the US. But it was great to come from a melodic instrument role into drums, because I had the "background knowledge" with me. If this is not your background, there are other paths to it. I'm also not saying you have to be formally trained and read music to be musical -- it helps, but it can be done without. (I do suggest investigating it at some point, though, especially if you want to work for hire some day.)
One of the most constant comments I get on my playing is that I play well to the song, or that I have "great ears". While I do listen to the whole song as I play, my job is made easier by the fact that I have listened to an awful lot of music over the years from all sorts of genres and I understand a lot of the basic song structures that prevail in those genres. For example, one of the reasons blues jams are so popular and prevalent is because that style of music conforms to a couple of common structures in about 95% of songs, which allows for an experienced set of players to almost read each others' minds, as they know where the I, IV, and V are going to be based on the very first runthrough in the song intro. They then get to play with dynamics, tension, transitions, extended solos, and even call-and-answer segments.
I recommend looking deeper into the music you listen to and hear, and becoming familiar with at least the basics of song structure and music theory (especially rhythm -- learning to divide the beat into quarter notes and beyond, as Larry mentioned).
Another part of musicality comes from getting the best sound out of your instrument. Learning to tune correctly, how to adjust tuning for rooms, how to get a good miked sound, how to correctly hit a drum to get the best sound, how to select good-sounding cymbals and play them with full tone and response -- these are critical things that will make you sound professional from the first beat you drop.
Finally, listen and adjust based on the other parts of the band. Sometimes you need to crank it up - or down - based on the other players. Support the soloist, the vocalist, and everyone else. You don't need to do a fill every four beats. The bass player might have something cool he's supposed to do that your fill walks all over. The guitarist might have a signature lick for that song he can't fit in if you're rolling down the toms. When you can hear and listen to what everyone else is doing, you stop being a bunch of individuals and become a team, mutually supporting each other. After all, there will be moments you get to shine too.