Learn Russian: Я не понимаю лекцию. - I do not understand the lecture.

The dictionary form is лекция (a feminine noun). Because it is the direct object receiving the action of the verb (the thing you don't understand), it must be in the Accusative case. For feminine nouns ending in -я, the ending changes to -ю.

I thought inanimate objects don't change their endings in the Accusative case?

That rule only applies to masculine and neuter nouns! Feminine nouns act differently: they always change their endings in the Accusative case, regardless of whether they are animate (like a person) or inanimate (like a lecture).

Why does the verb понимаю also end in -ю? Is it matching the noun?

No, it is just a coincidence! The verb ends in -ю because it matches the subject я (I). The dictionary form is понимать (to understand), and dropping the -ть and adding -ю is the standard way to conjugate this type of verb for the pronoun я.

Why does не come right before the verb?

In Russian, the negative particle не must always be placed immediately before the exact word you want to negate. To say that you do not understand, you simply put не right in front of понимаю.