Learn Russian: Учитель пишет статью о политике. - The teacher is writing an article about politics.

Only masculine and neuter inanimate objects stay the same as their dictionary form. Feminine nouns like статья (article) always change in the accusative case, whether they are alive or not! The ending -я changes to -ю.

The dictionary form of "to write" is писать. Why does it become пишет here?

The verb писать has a consonant mutation when conjugated. The с changes to an ш, making the stem пиш-. Then we add the standard "he/she/it" ending -ет to match учитель (the teacher). Be careful to remember this change, as pronouncing it with an "с" makes a completely different (and slightly embarrassing) word!

Why is the word for politics политике here instead of политика?

Because it comes right after the preposition о (about). This preposition always requires the Prepositional case. For feminine nouns ending in -а like политика, the ending changes to -е to show what the article is about.

Is учитель a masculine or feminine word?