Tempo - the speed of the beat. Fast, slow, moderate, etc.
Meter - the automatic breaking up of beats into patterns of two or three in our ear when we hear music. "How the beats are grouped".
Measure or "Bar" - when we break up the beats into patterns of two or three, a measure is one complete time through the patterns. Ex. 1 - 2, 1 - 2, 1 - 2 is three measures.

To determine the meter, find the beat first, then try to locate the downbeat (beat 1 of the pattern). How many beats until the next downbeat 1-2-3-4-1-2-3-4, etc.
4 beats is most common - really almost the same as 2 beats
3 beats is triple meter. Not nearly as common.
Happy Birthday, The Star-Spangled Banner, Break Away by Kelly Clarkson is in 6/8, End Of The Road by Boyz II Men, louis armstrong«s wonderful world, seal - kiss from a rose, Beatles, You've Got To Hide Your Love Away and Norwegian wood. Raining In Baltimore Counting Crows, the verses of "Lucy In The Sky With Diamonds", hallelujah by Shrek, George Michael one more try,
Nonmetric - when the beat is not evident.
Syncopation - an emphasis or accent placed in between the beats or on a weak beat. Gone by Kelly Clarkson

Rhythm - an organized and repeating pattern of beats and syncopations forming a musical idea. (I've modified this to " an organized and repeating pattern of shorts and longs forming a musical idea".)
Groove - the energy present in a piece of music that makes you bob your head, tap your feet, or dance down the hall.
Metronome - a clicker that keeps a steady beat. Used for practice by musicians - sets tempo (speed)
A metronome tempo is identified by how many beats will occur in 60 seconds.
A metronome tempo of 60 is slow - one beat per second
A metronome tempo of 120 is moderate tempo - 2 beats per second
Downbeat - beat 1 of a measure
In most commercial music, you can expect a snare drum on beats 2 and 4. This is called the "Backbeat"
Backbeat - beats 2 and 4 where you will commonly hear the snare drum
Pickup note - occurs before the downbeat as in Happy Birthday To You
Sometimes there will be a "fill" or buildup on beat 4 to set up the return of the downbeat