
egal
indifferent
Indifferent or unimportant; of no consequence. Usually used predicatively.
Mir ist egal, ob wir ins Kino oder ins Theater gehen.
It doesn't matter to me whether we go to the cinema or the theater.
indifferent / no matter
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indifferent
Indifferent or unimportant; of no consequence. Usually used predicatively.
Mir ist egal, ob wir ins Kino oder ins Theater gehen.
It doesn't matter to me whether we go to the cinema or the theater.

no matter whether/what/how
Introduces a subordinate clause and means 'regardless of whether/what/how...'.
Egal, was passiert, ich werde immer zu dir halten.
No matter what happens, I will always stand by you.

doesn't matter
A colloquial answer used to express that something is indifferent to you or that you have no preference.
„Möchtest du Tee oder Kaffee?“ – „Egal, ich trinke beides gern.“
'Would you like tea or coffee?' — 'Doesn't matter; I like drinking both.'
The word 'egal' is almost only used predicatively (e.g., 'Das ist egal' — That doesn't matter) or adverbially. It is not declined like a normal adjective before a noun.
Die Farbe ist mir egal.
The color doesn't matter to me.
To express to whom something is indifferent, the construction 'jemandem egal sein' is used. The person experiencing the indifference is in the dative.
Ihm ist es völlig egal, was andere über ihn denken.
He doesn't care at all what others think of him.
In the combination 'egal, ob/was/wer…' (no matter whether/what/who...), 'egal' introduces a subordinate clause. The main clause that follows usually begins with the subject (normal word order).
Egal, wie das Wetter wird, wir gehen spazieren.
No matter what the weather is like, we are going for a walk.
As a standalone answer, 'Egal!' is a very common, colloquial way to respond to a question and express that you have no preference.
„Was möchtest du essen?“ – „Egal, ich habe auf alles Appetit.“
'What would you like to eat?' — 'Doesn't matter; I'm in the mood for anything.'
immediate / same
Both words can express that something is unimportant to someone or makes no difference to them. In this meaning, Gleich typically takes a person in the dative and sounds somewhat more marked, whereas egal is more general and more common. Use Gleich in the fixed expression „jemandem gleich sein“; otherwise, use egal in most cases.

Welchen Platz wir bekommen, ist mir egal.
I don’t care which seat we get.
Welchen Platz wir bekommen, ist mir gleich.
I don’t mind which seat we get.