To concede something is to give in or stop protesting it. A loser concedes victory to the winner by surrendering. In a debate, one side may concede a point in the argument, or in other words, admit that the other side is correct in that point. The past tense of this word is conceded. It should be noted that the e in this word is pronounced with the long e sound. Also, this word is commonly misspelled and should not have two e‘s (as in succeed).
Conceit is a mass noun that means to be overly proud of one’s self and accomplishments. The adjective form is conceited. A good synonym for this word is vain. Note that this word is pronounced with ei making the long e sound.
While one is a verb and the other an adjective, the confusion of the two words comes from the similarity between t and d. The two sounds are made the same way in the mouth, but one is voiced (uses the vocal chords) and one is unvoiced.
Examples
Center-left challenger Isaac Herzog conceded defeat on Wednesday in Israel’s general election and said he had congratulated right-wing Prime Minister Benjamin Netanyahu on his win. [Business Insider]
The danger is that Tanner will turn into a priggish, conceited bore but, sporting a beard and still-dashing looks, the actor maximises the twinkling, good-humoured charisma of the man, while letting you appreciate his volatile mixture of self-certainty, cynicism and borderline hysteria. [The Telegraph]