Tempo is the speed of the underlying pulse of the music. If a piece is marked q = 80, then 80 beats per minute is the tempo.
Meter is the organization of the music into regularly occurring units. 3/4 is organized into units of three beats. 6/8, into units of six beats. The unit is expressed as a measure. Each meter has a readily identifiable "feel" -often with the first beat receiving more weight than the others.
Students get into difficulties sometimes when they try to practice slowly. We all need to do this, of course, it is a critical part of the learning process. But when playing the music slowly, they sometime fail to recognize the "feel" of the music at this new tempo and the rhythm is then easy to distort. Once distorted, it is all too easy to become accustomed to this new distorted version of the music without realizing it, and become habituated to it. This ends up being a signal challenge: playing your music at various tempi, always while maintaining a strong sense of the metric feel, of the "groove."
Groove is actually a pretty deep concept. Good percussionists have it. Musicians who play strongly rhythmic pop, rock, folk or jazz music have it. Certainly, denizens of Latin American popular music and flamenco have it. But what IS it?
First, it is a firm sense of tempo. In order to have the groove, you can't rush or drag the tempo. It has to be rock-solid. Working with a metronome, naturally, helps with this, but so does playing with others. Musicians with excellent groove are foot-tappers.
Consider the dotted-8th/16th-note grouping. This rhythm confounds most students. At it's simplest, it is an example of dividing a beat into four equal parts. It is absolutely critical that this can be done with precision, consistently. Only then, can subtle style-driven inflections be applied with certainty. Most students drag the rhythm toward a triplet feel. I call it the "Texas lope" (sorry, Texas), as it reminds me of the rhythm of riding slowly on horseback. In fact in many musical settings, style dictates a slight sharpening of the rhythm, approaching double-dotting. This is especially likely at slower tempi. Every rhythmic figure you encounter needs to be fully understood, and playable at various speeds.
When confronted with a complicated rhythm, it is always helpful to break it down. Write in the score where the beats are, where the half-beats fall. Break the measure into its sections (beats) or even smaller until you can suss-out the meaning of the rhythm. It is never good to put this off or hope it will get better with time. It will get better when it is analyzed and understood. Beware of learning rhythm from the performances of others. First, they may play it incorrectly, and second, they may be applying rubato or other inflections that you may distort in translation. You are never too old to count out loud.
It is immensely helpful to play with a mute. There is a new product called a "tremolo mute" (available at Strings By Mail), or simply take a (clean) sock or (dry) washcloth and roll it, insert it under the strings and slide it towards the bridge, as snugly as possible. If you've never done this before, the effect is revelatory. If you haven't done it for a while, try it again. By suppressing the ringing of the notes, you can suddenly hear your attacks much more clearly. You become a percussion instrument. If your arpeggio is uneven or if there's a hiccup in your tremolo, it will be immediately obvious. If you play your music with a mute and your eyes closed, just really listening to your rhythm, you'll easily find where improvements will help.
I spend quite some time thinking through my music in my mind. I hear the rhythmic flow of the music, that is, the rhythmic phrasing. I often use verbal aids like tak-i-taki-tak, taketa taketa ta, etc. If the rhythmic sense of a phrase is right on target, the music usually sounds pretty good. We forget how critical the rhythm is in conveying musical gesture. (Of course, then add dynamics, articulation, timbre, etc., etc., and you're in business!)
Much music is inflected in performance with subtle (and not-so-subtle) deviations from written rhythm and tempo. We refer to this as rubato. Rubato is critical and certainly a part of this discussion. But it will receive separate consideration in another post.
The most obvious marker for musical immaturity is poor rhythm. Make rhythm your most comfortable and confident element and the rest will be easy.