What is the interest rate that on the vertical axis of money markets? The money market diagram showing liquidity preference and money supply intersect at an equilibrium level of interest rate, but what interest rate is it? Is it the base rate (I presume this is the rate at which commercial banks lend out loans to borrowers) or FED Funds Rate/LIBOR? Or does it simply refer to interest rates in general?
The thing to keep in mind is that these diagrams refer to simple teaching economic models. These models do not necessarily reflect the complexity of the real world.


  The money market diagram showing liquidity preference and money supply
  intersect at an equilibrium level of interest rate, but what interest
  rate is it?


Within the teaching model, there is a portfolio allocation between "money" and some interest-paying instrument (labelled "bonds" or "bills"). The interest rate is the interest rate paid on that hypothetical instrument.

The usual interpretation is that this the short-term "risk-free" rate of interest. (Risk-free implies is that they are assumed to be free of default risk.) In the United States (for example), there are a number of such instruments: Treasury bills/bonds, the fed funds rate (domestic interbank rate), OIS, repo, interest-bearing reserves, etc. These instruments have interest rates that are generally close to each other. 

Simplified models that consist of a pair of intersecting lines will not be able to deal with the fact that such instruments have a variety of maturity dates (from overnight to 30 years).

The general assumption when discussing these models is that private sector borrowing rates will remain close to this risk-free curve. The fact that this is not always true is another part of reality they cannot deal with. That said, credit spreads for high quality borrowers tends to be less volatile than the level of the risk-free rate, and so the risk-free rate does roughly capture the general tendency of interest rates.


  Is it the base rate (I presume this is the rate at which commercial
  banks lend out loans to borrowers)


The only "base rate" I can think of is a rate associated with central bank dealings with banks. The prime rate is a benchmark borrowing rate set by banks for dealing with customers. For example, people have loan interest rates that contractually are set at "prime + x%".


  FED Funds Rate/LIBOR?


FED Funds represents reserve trading between domestic banks; it is considered a risk-free rate (no credit premium).

LIBOR is a rate for unsecured lending between international banks (in London). It has a credit risk premium embedded in it.