Learn Italian: Ho un orologio di plastica.

In Italian, you generally cannot use a material noun as a describing word (adjective) the way you do in English. To say what something is made of, you use di (of) followed by the material. So, you literally say "a watch of plastic" (un orologio di plastica).

Does orologio mean "watch" or "clock"?

It means both! Italian uses the word orologio for almost any timepiece, whether it is a watch you wear on your wrist or a clock on the wall.

Why is the article un instead of uno or un'?

Orologio is a masculine noun. For masculine nouns that start with a vowel, the correct word for "a" or "an" is simply un. The version with an apostrophe (un') is only used for feminine nouns that start with a vowel, while uno is for masculine words starting with specific sounds like a "z" or an "s" followed by another consonant.

Why isn't the sentence Io ho un orologio?

You can include io (I), but in Italian, it is completely normal to drop the subject word. The verb form ho can only mean "I have," so who is doing the action is already perfectly understood without it!