Tavaly kezdődött a tanfolyam.

Tavaly means last year.

It is a complete time adverb by itself, so you do not need any extra word or ending with it. It refers to the previous year relative to the time you are talking from.

A useful contrast:

• tavaly = last year
• egy éve = a year ago / for a year

So Tavaly kezdődött a tanfolyam means the course began in the previous year, not necessarily exactly one year ago.

The dictionary form is kezdődik, which means to begin / to start.

So:

• kezdődik = begins / is beginning
• kezdődött = began / started

It is not a passive form. Hungarian usually does not express this idea with a passive the way English sometimes can. Instead, Hungarian uses an intransitive verb here: the course itself began.

The form kezdődött shows past tense, in third person singular.

A simple way to think about it:

• kezdődik = it begins
• kezdődött = it began

In Hungarian, past tense is often marked with -t, but the actual form can appear as -tt, -ott, -ett, or -ött, depending on sound patterns. Here the result is -ődött.

Also, in third person singular, there is no separate word for it. The verb form already gives you began.

Because Hungarian uses two different verbs here:

• kezdeni = to start something
• kezdődni / kezdődik = to begin / to start

So:

• A tanár kezdte a tanfolyamot. = The teacher started the course.
• A tanfolyam kezdődött. = The course started / began.

English often uses start for both ideas, but Hungarian usually separates them.

Yes, that is also possible.

• kezdődött = began / was beginning
• elkezdődött = started / got started

The prefix el- often gives the sense that the action actually got underway. In many everyday contexts, kezdődött and elkezdődött are both natural, though elkezdődött can feel a little more explicit about the beginning happening.

So both of these work:

• Tavaly kezdődött a tanfolyam.
• Tavaly elkezdődött a tanfolyam.

A is the definite article the.

So:

• a tanfolyam = the course

Hungarian has two forms of the definite article:

• a before consonant sounds
• az before vowel sounds

Because tanfolyam starts with t, you use a, not az.

Yes. A tanfolyam is the subject.

Hungarian word order is much more flexible than English word order, so the subject does not always have to come before the verb.

Here:

• Tavaly = time expression
• kezdődött = verb
• a tanfolyam = subject

Also, tanfolyam has no special case ending here because it is in the basic form, which is what subjects normally use.

Yes, both are correct, but the emphasis is a little different.

• Tavaly kezdődött a tanfolyam.
  This starts with the time frame: last year. It is very natural if you want to say when it began.

• A tanfolyam tavaly kezdődött.
  This starts with the course, so it feels more like as for the course, it started last year.

Hungarian word order often reflects what is already known and what is being highlighted, not just grammar. So both versions are grammatical, but they organize the information differently.