Is there an implicit "unit of time" involved in electricity measurement? (e.g. miles per hour, kb per second, amps per ???)
Amps includes time...

Amps = Coulombs per second

That says more simply that...

Current = amount of charge per time interval

It's a flow rate metric. Like water... liters (volume --> amount) per minute (time)

In more depth


  In practical terms, the ampere is a measure of the amount of electric
  charge passing a point in an electric circuit per unit time with 6.241
  × 1018 electrons, or one coulomb per second constituting one ampere.


--Wikipedia Article

Probing


  When I touch the leads of my multimeter to a wire, what exactly is it
  "looking at"?


If you are in the voltage measurement mode, you are effectively measuring the "pressure" between the two leads -- the degree to which charges in one lead seek to reach the other (but can't). The reason the charge gradient can't be neutralized depends on the circuit. In a capacitor, for example, a barrier of some kind prevents it. The existence of a voltage between two points requires that such a gradient exists.

If you are in a current measurement mode, the leads are installed in the current path (in series with) and the meter is measuring how much charge flows through them in unit time (it actually does this indirectly by applying Ohm's law).

Further reading

Bodanis, David (2005), Electric Universe, New York: Three Rivers Press, ISBN 978-0‐307‐33598‐2