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LingakuLingaku[verb て-form] + もいい [i-adj 〜い → 〜くて] + もいい [na-adj] + でもいい [noun] + でもいい
てもいい gives or asks for permission: "you may", "it's okay to". It goes after a verb's て-form, and with か on the end it asks "may I?".
Literally it is "even if you do it, it's good": the て-form, も "even", and いい "good". The verb has to be in the て-form.
To say yes, reply with どうぞ or いいですよ. Answering with てもいいです sounds like someone in charge handing out permission, so keep that for when you are. To say no, use てはいけません, or more gently, trail off.
These go beyond asking and giving permission.
Among friends も often drops out.
A noun or な-adjective takes でもいい, and an い-adjective takes くてもいい. The meaning moves from permission to "that will do".
About your own action, てもいい offers to do it: "I don't mind ~ing".