Learn Turkish: Bir an bekliyorum. - I am waiting a moment.

In Turkish, you usually don't need to use the pronoun ben (I) because the personal ending on the verb already tells us who is doing the action. Here, the -um at the very end of bekliyorum means 'I am doing it'.

The dictionary form of the verb 'to wait' is beklemek. Why is it spelled bekliyorum here instead of bekleyorum?

The root of the verb is bekle-. When you add the present continuous ending -iyor (which shows an action happening right now), a special spelling rule kicks in. If a verb root ends in the vowels a or e, that vowel narrows to fit the -iyor ending. So, the e at the end of bekle- changes into an i, giving us bekliyorum.

Why isn't there a word for 'for' in this sentence, like 'waiting for a moment'?

In English, we often use 'for' to show how long an action takes. In Turkish, you don't need an extra preposition like that for time durations. You just state the amount of time directly into the sentence. The phrase bir an (a moment) works perfectly on its own to mean 'for a moment'.