A groove is usually defined by how certain notes relate to time. It isn't "everything being consistently late/early" or anything like that. Since no one else hears the click or sees the grid, the relationship to that grid cannot have an effect on the groove.
Since there are many different grooves, there are many different answers as to what a good one will look like. Sometimes the time keeping instrument (normally that hat or ride) will push or pull certain hits in a measure. For example lagging slightly on '3' in a 4/4 bar. Sometimes the time keeping instrument is consistent, but the kick pushes and the snare lags compared to the hat...or the other way around.
Once you have the drums doing the groove, then the other instruments do the same thing. A bass note being slightly early will have a different feel to one being slightly late or on time. Oftentimes, a good groove will have some notes in the line consistently late and some consistently early.
The problem comes in when you try to manufacture it. It is so much easier to play a groove than it is to define it and create it on purpose. And, quite frankly, your job as a musician is to play the groove. This takes practice and experience.
What happens a lot now is that people spend more time editing their parts than practicing and performing them. Which means that we are ending up with so-so musicians who are excellent audio editors. Nothing wrong with being an editor, but what is it you are trying to be? A musician or an editor?