Sed magistra monet cladem unius populi etiam iis qui vicerunt dolori esse posse.

The sentence breaks down like this:

• Sed magistra monet = But the teacher warns / advises / points out
• cladem unius populi etiam iis qui vicerunt dolori esse posse = the idea being reported

So the whole sentence consists of:

• a main clause: Sed magistra monet
• an indirect statement after monet

That indirect statement is a very common Latin construction.

Latin often does not use a separate word meaning that after verbs of saying, thinking, knowing, hearing, and similar ideas.

Instead, it often uses an accusative + infinitive construction, usually called an indirect statement.

Here:

• cladem is the accusative
• esse posse is the infinitive phrase

So:

• magistra monet
• literally: the teacher warns [the disaster of one people to be able to be a grief even to those who conquered]
• naturally in English: the teacher warns that the disaster of one people can be a grief even to those who won

Because in an indirect statement, the subject of the infinitive is put in the accusative.

The dictionary form is clades = disaster, defeat, calamity.
Its accusative singular is cladem.

So in this sentence, cladem is not the direct object of monet in the ordinary sense. It is the subject of esse posse inside the indirect statement.

In other words:

• cladem ... esse posse = that the disaster ... can be

Because posse means to be able or can, and it often takes another infinitive with it.

So:

• esse = to be
• posse = to be able

Together:

• esse posse = to be able to be
• more natural English: can be

So dolori esse posse means can be a source of grief or can be painful.

Dolori is dative singular of dolor, meaning grief, pain, or sorrow.

Here it is part of a very common Latin pattern:

• dolori esse alicui

literally:

• to be for grief to someone

more natural English:

• to be a grief to someone
• to cause someone grief
• to be painful to someone

So dolori is not just a random dative. It is part of an idiomatic construction.

This is a classic double dative construction.

The two datives do different jobs:

• dolori = dative of purpose/result
  (as a grief, for grief)

• iis = dative of reference / the person affected
  (to those people)

So:

• iis ... dolori esse
  = to be a grief to those people

This is a very Latin way of expressing an idea that English usually handles differently.

Iis is the dative plural of is, ea, id, meaning to those or for those.

It is dative because it goes with the construction just discussed:

• dolori esse iis = to be a grief to those people

You may also see this form spelled eis. Both spellings are standard.

So:

• etiam iis = even to those

Qui vicerunt is a relative clause modifying iis.

• iis = those
• qui = who
• vicerunt = won / were victorious / have conquered

So:

• iis qui vicerunt = to those who won

This is very common Latin: a pronoun like iis is followed by a relative clause that explains which people are meant.

Because qui is the subject of vicerunt.

Inside its own clause:

• qui vicerunt = who won

Since who is doing the winning, it must be nominative.

This is an important point: the case of a relative pronoun depends on its role inside its own clause, not on the case of its antecedent.

So even though iis is dative, qui is nominative because it is the subject of vicerunt.

Vicerunt is perfect active indicative, 3rd person plural, from vinco = conquer, defeat, win.

Depending on context, the Latin perfect can be translated in more than one way:

• they won
• they have won
• sometimes they were victorious

Here the most natural English is usually:

• those who won
• or those who were victorious

It is a genitive phrase depending on cladem.

• cladem = the disaster
• unius populi = of one people / of one nation

So:

• cladem unius populi = the disaster of one people

Here populus means a people in the sense of a nation or group, not just people in a general mass sense.

Because unius is the genitive singular of unus.

The adjective unus, una, unum is a little irregular. Its genitive is:

• unius = of one

So:

• unus populus = one people / one nation
• unius populi = of one people / nation

Since the phrase means of one people, the genitive is required.

Etiam means also, even, or still.

Here it most naturally goes with iis qui vicerunt:

• etiam iis qui vicerunt = even to those who won

So the idea is that the disaster of one people can cause grief not only to the defeated, but even to the victors.

It can mean warns, but depending on context it may also be understood as:

• reminds
• advises
• points out
• teaches

The core idea is that the teacher is calling attention to something important.

So magistra monet could be rendered in slightly different ways in English, depending on the tone of the passage. Grammatically, though, it is still the verb that introduces the indirect statement.

Yes.

A very literal unpacking would be:

• But the teacher warns [the disaster of one people] [even to those who won] [to be able to be] [for grief].

A more grammatical English unpacking:

• But the teacher warns that the disaster of one people can be a grief even to those who won.

And a smoother English version might be:

• But the teacher warns that the disaster of one nation can bring grief even to the victors.