In Japanese it’s all a matter of style. Whenever the language is used, a basic choice needs to be made between the formal way of speech, known as 敬体 (keitai, distal style), and the corresponding mode for informal speech, called 常体 (jōtai, direct style).
The basic feature of the distal style is that sentences end in either です (desu), the “be” copula that comes with nouns and adjectives, or ます (masu), the concurring ending for verbs. As for the latter, each and every Japanese verb has a plain form, ending in (r)u, from which a corresponding masu form is directly derivable. So much for the easy part.
The main problem is the relationship between the formal copula desu and its informal version da, which on closer inspection are not so, well, corresponding, at all. As said, these are used mainly when sentences end in nouns or adjectives. What makes them so tricky is that there are actually two types of adjectives in Japanese, and these require different treatment.