Actually, 4/8 at 60bpm could be written as 4/4 ALSO at 60bpm.
Meters (what the time signature denotes) have two properties: whether they are simple or compound, and whether they are duple, triple, or quadruple (though some argue it is impossible to aurally tell the difference between duple and quadruple). The first property refers to how the beat is DIVIDED - simple beats are divided in 2, and compound beats are divided in 3. The second refers to how many beats are in a measure.
Marches are nearly always in simple duple (usually 120bpm), meaning they have two beats in a measure, and the beats are divided in two. They can be written in 2/4, 2/8, 2/2 (cut time) etc. Semper Fidelis is an example of a march in COMPOUND duple - 6/8. The dotted quarter note gets the beat, and there are two beats per measure, but the beats are divided into three.
In simple time, the possible top numbers of a time signature are 2 for duple, 3 for triple, and 4 for quadruple. The bottom number represents the beat note. In compound time, the possible top numbers are 6 for duple, 9 for triple, and 12 for quadruple. The bottom number represents the DIVISION of the beat (the beat of 6/8 is actually 3 eighth notes - a dotted quarter note).
Now, this explanation is based on the Common Practice Period (1600-1900) standards. Modern composers sometimes break these conventions and even use irregular time signatures. But this is still very useful knowledge.
As for tempo, it is simply how fast the composer wants the perceived beats to go by. As for whether he uses quarter notes, eighth notes, half notes, 128th notes or whatever for the beat or beat division, I think that is simply up to whatever he wants it to look like. :)