Я наконец понял то, что учитель объяснил на лекции.

Понял is the past tense of the perfective verb понять.

Here it means something like I finally understood / I finally got. The speaker is talking about a completed moment of understanding in the past.

Compare:

• Я понимаю = I understand / I am understanding now
• Я понимал = I understood / I used to understand / I was understanding (imperfective, focusing on process or ongoing state)
• Я понял = I understood / I realized / I got it (perfective, focusing on the result)

Because наконец means finally, Russian naturally uses понял here: the moment of understanding was achieved.

Наконец means finally / at last.

In this sentence, it emphasizes that the speaker understood something only after some delay or difficulty:

• Я наконец понял... = I finally understood...

Its position is fairly flexible, though some positions sound more natural than others:

• Я наконец понял то, что учитель объяснил на лекции.
• Наконец я понял то, что учитель объяснил на лекции.

Both are possible. Putting наконец right before понял is very natural because it highlights the verb: finally understood.

This is a very common Russian pattern:

• то, что = that which / what / the thing that

So:

• Я понял то, что учитель объяснил на лекции. literally resembles:
• I understood that which the teacher explained in the lecture.

In natural English, we usually just say:

• I understood what the teacher explained during the lecture.

Russian often uses то, что where English simply uses what.

This is an important difference.

1. Я наконец понял то, что учитель объяснил на лекции.
   = I finally understood what the teacher explained in the lecture.
   Here, the speaker understood the content of the explanation.

2. Я наконец понял, что учитель объяснил на лекции.
   This sounds like I finally realized that the teacher explained it in the lecture or I finally understood that the teacher did explain it during the lecture.
   Here, что introduces a full that-clause, not what.

So то, что is needed if you mean what the teacher explained, not that the teacher explained.

То is the pronoun то (that / that thing) in the accusative singular neuter form.
In this sentence, it is the direct object of понял:

• понял что? → то

Then что учитель объяснил на лекции is a subordinate clause that explains what то refers to.

So the structure is:

• Я понял [то], [что учитель объяснил на лекции].

You can think of что here as a relative word meaning something like which / that / what depending on how English would express it.

Because что учитель объяснил на лекции is a subordinate clause.

Russian normally separates subordinate clauses with commas:

• Я понял то, что...
• Я знаю, что...
• Я думаю, что...

So the comma is required here.

Because учитель is the subject of the subordinate clause:

• учитель объяснил = the teacher explained

Subjects are normally in the nominative case, and the nominative singular form is учитель.

Объяснил is the perfective past tense of объяснить, meaning explained as a completed action.

That fits well here because the teacher gave an explanation, and the speaker is referring to that finished explanation.

Compare:

• учитель объяснил = the teacher explained / finished explaining
• учитель объяснял = the teacher was explaining / used to explain

In this sentence, the completed explanation is more natural.

Russian uses на лекции to mean at the lecture or during the lecture.

This is just the normal idiomatic choice:

• на уроке = in class / during the lesson
• на лекции = at the lecture / during the lecture
• на собрании = at the meeting

Even though English often uses in, Russian frequently uses на for events or organized activities.

Because after на in this meaning, Russian uses the prepositional case:

• лекция → на лекции

So:

• лекция = lecture
• на лекции = at the lecture / during the lecture

This is a regular case change.

Yes. Russian word order is more flexible than English, although different orders change emphasis.

The original sentence is neutral and natural:

• Я наконец понял то, что учитель объяснил на лекции.

Other possible versions include:

• То, что учитель объяснил на лекции, я наконец понял.
• Я понял наконец то, что учитель объяснил на лекции.

These are grammatically possible, but the original version is the most straightforward for ordinary speech.

Only the past-tense verb form changes:

• Я наконец поняла то, что учитель объяснил на лекции.

Russian past tense agrees with gender in the singular:

• понял = masculine
• поняла = feminine
• поняло = neuter
• поняли = plural

So a woman would say поняла.

Sometimes in colloquial speech you may hear sentences without то, but with this meaning то, что is the clearest and most standard structure.

So this is best:

• Я наконец понял то, что учитель объяснил на лекции.

If you remove то:

• Я наконец понял, что учитель объяснил на лекции

the meaning usually shifts to I finally understood that the teacher explained it in the lecture, not I finally understood what the teacher explained.

So in this sentence, то is important.

Russian has no articles, so it does not mark a / an / the directly.

That means:

• учитель can mean a teacher or the teacher
• лекция can mean a lecture or the lecture

The exact meaning comes from context.

In your sentence, English naturally uses the teacher and the lecture because the situation is already known or specific, but Russian does not need separate words for that.