Learn Hardcore Portuguese (Brazil): O julgamento começa hoje.

Portuguese often uses the definite article more than English does. In this sentence, o julgamento means the trial / the judgment as a specific event that both speaker and listener can identify.

Leaving out the article is sometimes possible in Portuguese, but here o julgamento sounds the most natural.

O julgamento começa hoje. = a specific trial starts today.
Julgamento começa hoje can appear in headlines, where articles are often omitted.

Why is julgamento masculine?

Because julgamento is a masculine noun, so it takes o.

A useful pattern is that many nouns ending in -mento are masculine:

o julgamento
o documento
o momento
o sentimento

There is no absolute rule that works for every noun in Portuguese, but -mento is a very reliable masculine ending.

Why is it começa and not começo?

Because the subject is o julgamento, which is third person singular: it begins/starts.

The verb começar in the present tense goes like this:

eu começo
você / ele / ela começa
nós começamos
vocês / eles / elas começam

So:

Eu começo hoje. = I start today.
O julgamento começa hoje. = The trial starts today.