Tempo is the speed you play at and Rhythm is the beat.
Rhythms name the type of tune - Reel, Jig, Mazurka, Waltz, etc.
You can have slow jigs, fast jigs, slow reels and fast reels slow waltzs and fast waltzs. Each ryhthm can be played more slowly or more quickly by different players and in different circumstances.
If the music is meant for dance there is usually a comfortable tempo for the dancers, otherwise anything goes.
In folk music, there are several basic rhythms. Typically, the rhythms go like this:
|Polka||2/2||1 and 2 and 1 and 2 ...|
|Waltz||3/4||1, 2, 3, 1, 2, 3 ....|
|Reel||4/4||1,2, 3, 4, 1, 2, 3,4 ...|
Bold means the beat is emphasised Italic means the beat is emphasised but less strongly A dash (-) means the note is held longer and the next one is shorter, giving a kind of 'bounce'
Note: The Reel and Jig in Irish music are played differently from the Jig and Reel in English music although they share the timing. Indeed the Irish Jigs and Reels are played differently in different parts of Ireland - variations in speed and 'bounce' especially.
Yes, there's an obvious joke there somewhere.
The thing about music is that it does have a beat and you can count the beats. In a time signature (like 4/4) the first number is the number of beats you count (1,2,3,4 and repeat). How do you know when to repeat? The beats should be emphasised and we're straying into the realms of dancing here. The second number is the typical length of the note (used to be called Semibreves, minims, crotchets,quavers and so on but now called whole note, half note, quarter note, eighth note etc)
The time signature is the number of beats in a bar over the default note length. So 4/4 just means 4 beats and each one's a crotchet, 6/8 means 6 beats and each one's a quaver.
Although the default note length is given in the time signature, the actual note lengths can be different, as long as it all adds up to the right amount per bar. A reel in 4/4 doesn't have to have 4 crotchets in a bar (4 x a quarter = 1) - it can have 8 quavers (8 x an eighth = 1) or a mixture as long as it adds up.
This gives tunes their wonderful diverse quality and leads to effects such as syncopation.