Learn German: Peters Geschenk ist wunderschön.

In German, when you show possession with a person's name, you simply add an -s directly to the name with no apostrophe. So "Peter's" becomes Peters. You only use an apostrophe in German if the name already ends in an s-sound, like Max or Jonas.

Why is there no word for "the" or "a" before Geschenk?

Just like in English, a possessive name takes the place of the article. You would say das Geschenk (the gift), but when you state the owner, it simply becomes Peters Geschenk (Peter's gift) without any extra article.

Does the gender of Geschenk change how we say Peters?

No, it doesn't! Geschenk is a neuter noun (das Geschenk), but the -s ending on Peters is just there to show that Peter owns it. It stays Peters whether the following noun is masculine, feminine, or neuter.

What is the difference between schön and wunderschön?

schön simply means "beautiful" or "nice". Adding wunder- (which comes from das Wunder, meaning "miracle" or "wonder") makes the adjective much stronger. wunderschön translates to "gorgeous" or "stunning".