Modality refers to a set of related concepts primarily involving the attitude of the speaker of a sentence towards the reality of a particular assertion. What exactly that means is complicated and best illustrated with an example:
(12a) Tad programs computers for a living.
(12b) Tad must program computers for a living.
In sentence (12a), the speaker asserts the truth of a proposition. In (12b), by contrast, the speaker qualifies the proposition. The situation is presented not as one the speaker knows directly but as one the speaker has inferred. In other words, in (12b), must indicates something about the speaker's mental state. These sentences, therefore, contrast in their modality.
Mood refers to a grammatical system that is primarily used to convey modality. The difference between mood and modality is parallel to the difference between tense and time. Like time, modality is a semantic concept; like tense, mood is a grammatical realization of a concept. For the most part, English expresses mood analytically, through a system of modal auxiliaries. As with tense, mood does not always correspond in a simple fashion with modality. One modal verb can express several different modalities, depending on the context. And just as time can be expressed in different parts of a sentence, for example by prepositional phrases, modality can be indicated with things other than auxiliary verbs:
(13) I heard his supposed apology.
In sentence (13) the adjective supposed expresses the speaker's conclusion that the apology is not a valid one, for example because it lacks sincerity. Words such as supposed, then, express modality, but not mood.
Sentence (12a) represents the default situation, one without a modal verb, in which the speaker simply indicates that something is true. This unmarked situation is called the indicative mood, although since this is the ordinary case, we usually don't mention it unless we're contrasting it with another mood.
In some grammar books, the presence of a modal auxiliary is said to mark the conditional mood. This label reflects the fact that modal auxiliaries commonly appear in sentences that express a condition:
(14) If you build it, they will come.
However, the label conditional is not ideal. There are many other situations in which modal auxiliaries appear other than the conditional structure. Further, many conditional sentences do not use modal auxiliaries:
(15) If he got a ticket to the concert, he was lucky.
Because modal auxiliaries express a variety of different modalities, we will not try to lump them all into a single mood. Instead, we will simply call such verb phrases modal, and if we need to distinguish among them, we will do so by their meaning.
 Exceptions to the analytical nature of English mood are the constructions traditionally called the "subjunctive", which are marked on the verb itself. They play a fairly small role in the grammar of English, but are more prominent in languages like Spanish, French, or Latin.