Learn German: Ich möchte ein Bier und ein Wasser, bitte. - I would like a beer and a water, please.

möchte translates to "would like" and is the polite way to ask for things. Just like in English, saying "I would like a beer" (Ich möchte...) is much more polite for ordering at a restaurant than demanding "I want a beer" (Ich will...).

I learned that modal verbs push the action verb to the end. Where is the second verb in this sentence?

Normally, you would use an action verb at the end, like haben (to have) or trinken (to drink). However, in German, if the context is obvious, you can completely drop the second verb! You can just say Ich möchte plus the thing you want to order.

Why are we using ein for both drinks?

Both Bier and Wasser are neuter nouns (das Bier, das Wasser). Because you are asking for them, they are the direct objects of the sentence (this is called the accusative case). For neuter nouns, the word for "a" is ein whether it is the subject or the object, so the spelling doesn't change here.

Can I put bitte somewhere else in the sentence?

Yes, you could put it at the beginning (Bitte, ich möchte...), but tacking it onto the very end with a comma is the most natural and common way to politely finish a request in spoken German.