Learn German: Ich habe gestern einen Platz reserviert.

Verbs that end in -ieren (like reservieren, studieren, or fotografieren) do not get a ge- added to the beginning in the past tense. You just take the verb stem (reservier-) and add a -t at the end.

Why is reserviert at the very end of the sentence?

In the German spoken past tense (the Perfekt), the helper verb (habe) takes the normal second position in the sentence, while the past participle (reserviert) gets pushed to the very end. This creates a frame around the rest of the sentence.

Why is it einen Platz and not ein Platz?

Platz is a masculine noun (der Platz). Because it is the direct object of the sentence (the thing being reserved), it must be in the accusative case. For masculine nouns, ein changes to einen in the accusative.

Could I change the word order and say 'Ich habe einen Platz gestern reserviert'?

You will be understood, but it sounds unnatural. German strongly prefers putting time words like gestern (yesterday) before direct objects like einen Platz. A handy rule of thumb is Time before Object.