だ is called a copula, roughly translating to the English verb "to be" (is, are, am, etc.). It is not a particle but a suffix that attaches to the end of nouns and adjectival nouns (na-adjectives). It is used in informal conversation, as opposed to its polite counterpart です, which it seems you already know. Like です, its basic use is to equate two things as equal; in other words, to say one thing is another thing.
私は学生だ - I am a student.
彼は大丈夫だ - He is all right
As I stated somewhere in the first paragraph, you can only attach it to nouns and na-adjectives (which are nouns that behave like adjectives). You cannot attach it to i-adjectives; they stand on their own at the end in informal speech. This is where だ differs from です, which attaches to i-adjectives.
Also unlike です, which only really appears at the end of the main clause, だ often appears at the end of quotations and subordinate clauses (but not relative clauses), even if the whole sentence is polite.
元気だと言った - He said he was fine.
冬だから客もあまりいません - Because it's winter, there aren't many customers.
The negative of だ is じゃない, which now, just like the negative form of verbs, conjugates like an i-adjective (so the negative past is じゃなかった, the -て form is じゃなくて, etc.).
私は学生じゃない - I am not a student.
友達じゃない - (He) isn't my friend.
全然好きじゃなかった - (I) didn't like it at all.
Grammatically speaking, だ and じゃない are always required. You can't end a sentence with a noun.
However, since it's informal speech, this rule is frequently ignored.
When used as a sentence-ending particle, の can generally do one of two things to a sentence, neither of which have the same function as だ. Firstly, it can emphasize emotion. This usage is used mostly by women or children.
ブロッコリーが好きじゃないの - I don't like broccoli
Secondly, it can be used as an explanation, or in other words, to fill in an "information deficit". This の is basically the same thing as のだ/んだ, but with the だ dropped. See more information about the のだ/んだ construction and meaning here or here. の as a question marker applies to this as well- it is the short and informal form of のですか.
今は忙しいの - I am busy (as an explanation)
お腹が空いたの - I was hungry (as an explanation)
どこに行くの(ですか) - Where are you going?
It is a little more complicated than that, but the links above should be good at explaining what の/のだ is used for as opposed to normal statements and questions.
Note, though, that sentence-ending の must be preceded by な if the preceding word is a noun so that it can be distinguished from the other uses of の.
私のだ - it is mine
私なのだ - it is me (as an explanation).