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.

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
'gleich' and 'egal' can both mean that something is not important. For example, you can say 'Das ist mir gleich' or 'Das ist mir egal' (It doesn't matter to me). 'egal' is very common in everyday language and sounds direct. 'gleich' in this meaning sounds a little calmer or less colloquial, but it is only used in fixed forms like 'mir gleich'. Learners may confuse the words because in this sense they are almost the same. The important point is: 'gleich' also has other meanings, for example 'right away' or 'same'. 'egal' is used more when you have no preference or consider something unimportant.

Mir ist egal, welchen Film wir heute Abend sehen.
I don't care which film we watch tonight.
Ob wir am Samstag oder am Sonntag fahren, ist mir gleich.
It doesn't matter to me whether we go on Saturday or on Sunday.