In a very general sense the interest rater is a number that tells us how much income can be earned by lending money, or equivalently how much cost is associated with borrowing money. The higher the r is, the more income one expects to earn by lending money. It is not possible to define an r mathematically so that it applies to all sorts of lending/borrowing operations. One should rather speak of an interest rate in relation to a specific lending/borrowing contract.
For example, the interest rate pertaining to certain short term deposits between banks is defined in a particular way and referred to as ibor rate The interest rate pertaining to interest rate swaps is defined in another way and referred to as swap rate Although not existing in the actual market, the interest rate pertaining to a theoretically conceivable instantaneous riskless deposit is defined appropriately and referred to as short rate
Given a specific contract C, the associated interest rater depends on the time t, when the value r is agreed between the parties entering into the contract C. We may therefore speak of a function r(.) that maps each time t to the respective interest rate value r(t). Assuming t = 0 designates the time now, the value r(t), t > 0 is not a simple number but rather a random variable, since it is not possible to know with certainty the interest rate that is going to prevail at the future tinme t. It follows, the function r(.) represents a mapping from t to some random variable, and therefore is a stochastic process. In the context of derivatives pricing, all what the so called "interest rate models" do, is to first choose a particular definition of r and then assume a particular mathematical form of the process r(t).
For example, in the so called "Market Model", either the ibor rate or the swap rate is chosen as the r and a lognormal diffusion is assumed for the respective process. On the other hand, in the so called "Short Rate Models", the short rate is chosen as the r and any out of a great variety of possible diffusions can be adopted for the respective process.