The word пойдёмте has two plural affixex -ём and -те. The reason is that the form пойдём is an indicative form used as an imperative one, but an imperative form without -те seems to be impolite when you are ‘на вы’ with your interlocutor(s).

The plural past indicative form пошли (not to be confused with its homonym пошли — imperative of послать ‘send’) is also used as an imperative. E.g. «Пошли в кино» “Let’s go to the cinema”. «Пошли отсюда!» “Go away!”.
And the form пошлите (with two plural affixex -и and -те) is also used in colloquial speech, but it is considered substandard, not a literary form.