p(D,H) is called the likelihood, and it is a forward probability.
p(H) is called the prior probability that hypothesis H is true, or just "prior" for short. It incorporates all a priori knowledge that you have. If you have a new nickel from the US mint and believe that nickels are manufactured with even weighting, you might set this close to or equal to 1. If you are playing a betting game with a known con artist / convicted criminal, you might set this close to 0.
What to do when you have no a priori information was controversial for over 200 years. (p(H) in this case is called an ignorance prior.) Laplace, who derived inverse probability independently of Bayes, said that you should assign equal probability to every hypothesis because there is no compelling reason to do otherwise. He called this the Principle of Insufficient Reason, and it was attacked so viciously and continuously that Bayesian inference was roundly discounted until roughly the 1970's or 80's. Physicist E. T. Jaynes finally showed that Laplace was right because the equal probability prior is the only choice that maximizes information entropy. There is still some small controversy over the correct scale-invariant form for an equal probability prior in certain problems, but this is unimportant here. For your problem, if it were known that the coin had to be either fair or crooked but you had absolutely no a priori information which, you would assign p(H)=1/2.
p(D) is the probability of observing the data, and it is often harder to find than p(H). In your problem, it is the probability of seeing H T independent of any coins or other parameters. In other words, it is the probability of seeing H T over the entire universe of coin tossing experiments with all possible coins (fair, bent, weighted, two-headed and two-tailed).
Basically, then, there is no simple answer to your question. You can fill in numbers according to your knowledge and beliefs, getting an answer that is consistent with them. It is often pointed out that Bayesian inference has parallels to human learning and decision behavior. If you acquire additional information, you update the prior (and possibly p(D)) to get improved estimates.
Problems are often reformulated as ratios to avoid the problem of finding p(D). In radar, one forms the ratio \Lambda=\frac{p(H_1,D)}{p(H_0,D)}, where H1 is the hypothesis that a received signal contains a weak echo in noise and H0 is that the signal is pure noise without an echo. Note that p(D), which is common to both, cancels from the ratio. If, in addition, p(H_1)=p(H_0)=0.5, (an ignorance prior) then \Lambda=\frac{p(D,H_1)}{p(D,H_0)} and this is called a Likelihood Ratio Test (LRT). and is widely used in science, engineering and statistics. A detection is declared when \Lambda\geq 0.5 .
