You have two fixed factors and one random factor. 'Drug A' (with levels "control" and "treatment") and 'Drug B' (with levels "dose 1", "dose 2" and "dose 3") are fixed factors. 'Rat' is random factor, nested within Drug A.
It is not always easy to decide whether a factor is fixed or random. If a factor is something that the experimentor has deliberately applied, is under their control, and the levels can be reproduced by someone else, then this is a fixed factor.
You have deliberately applied drug A, or not (the control group), and also deliberately given 3 specific doses of drug B to the rats. Someone else could follow your methodology and the meaning of drug A, drug B, and the timing and amounts of drug B would be the same in their experiment as it is in yours.
In contrast, each rat has an individual ID and is a level of the 'Rat' factor. Let's take the first rat in the treatment group, for example. No-one else can replicate that rat directly in their experiments, they would have to use a different rat.
Your conclusions from your experiment are about the physiological effects of drug A in combination with drug B in those specific conditions (fixed factors) on the population of rats in general (from which your test subjects are a sample of, and hence a random factor).
Sokal and Rohlf (1995, pp198-205) give a more detailed account of the difference between fixed and random factors with respect to Model I and Model II ANOVAs and with the consequences for interpretation and the formulas for expected Mean Squares.
The Minitab support pages http://support.minitab.com/en-us/minitab/17/topic-library/modeling-statistics/anova/anova-models/fixed-and-random-factors/ and The Analysis Factor website http://www.theanalysisfactor.com/specifying-fixed-and-random-factors-in-mixed-models/ also have some useful information regarding when to consider a factor as fixed or random. And, there are some more thoughts on this highly viewed CrossValidated question What is the difference between fixed effect, random effect and mixed effect models?.
Reference: Sokal RR and Rohlf FJ (1995) Biometry. The principles and practice of statistics in biological research. 3rd edition. WH Freeman & Company, New York.