Learn Icelandic: Þetta er hundur.

When you introduce something with the phrase "This is...", Icelandic always uses the fixed neuter word þetta. Think of it as meaning "This general thing here is...". The gender of the noun that follows doesn't change it!

Where is the word for "a" in this sentence?

Icelandic does not have a separate word for "a" or "an". You just use the bare noun. Because of this, hundur translates to either "dog" or "a dog" depending on what sounds natural in English.

How would I change this to say "This is the dog"?

You would add the definite suffix to the end of the noun. Because hundur is a masculine word, it takes the -inn ending. The sentence would become Þetta er hundurinn.

Is the -ur at the end of hundur a special ending?

Yes, -ur is a very common ending for masculine nouns in their basic dictionary form. When you add the "the" suffix, it just tacks right onto it: hundur + inn = hundurinn.