What exactly is a random disturbance in control theory? I'm embarrassed to even ask but... 

I frequently see this word used in articles about dynamical systems, but not until now have I questioned what it really is. I understand that it is opposite of a deterministic system, but again, not very explanatory.
A random disturbance or noise in a dynamical system is a result of the error in the modelling of the system or the error due to sensor. For e.g. imagine that you are trying to model a complex dynamical system. But due to its complexity you cannot model it exactly. So, there is some error in your modelling. If you know what could be the possible error in your modelling then you call that error as random disturbance or noise. This type of error is called modelling error.

Similarly, when you are using a sensor to read some data, the sensor may give you some error. For e.g, if you are using a vernier calliper to measure something, then there may be error due to offset of zero. Similarly, if you are using an electro-mechanical sensor, then there may be some error in the reading of current or voltage. All such errors are called random disturbance or noise. This type of error is called observation error.
A random disturbance or noise in a dynamical system is a result of the error in the modelling of the system or the error due to sensor. For e.g. imagine that you are trying to model a complex dynamical system. But due to its complexity you cannot model it exactly. So, there is some error in your modelling. If you know what could be the possible error in your modelling then you call that error as random disturbance or noise. This type of error is called modelling error.

Similarly, when you are using a sensor to read some data, the sensor may give you some error. For e.g, if you are using a vernier calliper to measure something, then there may be error due to offset of zero. Similarly, if you are using an electro-mechanical sensor, then there may be some error in the reading of current or voltage. All such errors are called random disturbance or noise. This type of error is called observation error.