Houd de kinderen in het oog.

Houd is the imperative form of houden, so it means keep, watch, or look after as a command or instruction.

So Houd de kinderen in het oog is telling someone to do that.

A useful comparison:

• Ik houd = I keep
• Houd! = Keep! / Watch!

In informal Dutch, you may also see hou instead of houd.

Because kinderen is plural, and in Dutch all plural nouns take de.

So:

• het kind = the child
• de kinderen = the children

This is very common in Dutch: a singular noun may be a het-word, but its plural still takes de.

Because in het oog houden is a fixed idiomatic expression. Dutch normally uses the singular here: the eye, not the eyes.

So even though English often says keep an eye on, Dutch says:

• iemand/iets in het oog houden

Literally, it looks like hold someone/something in the eye, but you should learn it as a set phrase meaning to keep an eye on or to watch carefully.

Because oog is a het-word.

So:

• het oog = the eye
• een oog = an eye

In the sentence, in het oog is just the normal article+noun combination inside the expression.

Because in Dutch, the finite verb comes first in an imperative sentence, and the rest of the verbal expression appears later.

Here, in het oog houden behaves like a fixed expression that gets split up:

• Houd = the verb part in first position
• de kinderen = the object
• in het oog = the rest of the expression

So the structure is very natural:

• Houd de kinderen in het oog.

You can think of it as similar to how separable verb-like expressions often work in Dutch.

It is mostly an idiomatic expression.

Word for word, it is something like:

• houden = to hold/keep
• in het oog = in the eye

But as a whole, iemand in het oog houden means:

• to keep an eye on someone
• to watch someone carefully
• to keep someone under observation

So it is better to learn the whole phrase together rather than translating each word separately.

Yes. Hou is a common variant of houd, especially in speech and informal writing.

So both are possible:

• Houd de kinderen in het oog.
• Hou de kinderen in het oog.

Generally:

• houd can feel a bit more standard or careful in writing
• hou is very common in everyday Dutch

It can be either. The Dutch imperative without a pronoun does not have a different form for singular and plural.

So Houd de kinderen in het oog could mean:

• You, keep an eye on the children
• You all, keep an eye on the children

The exact meaning depends on context.

Usually it means watch them / keep an eye on them / supervise them, not simply look at them.

So it suggests attention over a period of time:

• make sure they are okay
• notice what they are doing
• prevent trouble

It is more about monitoring than just staring.

They are very close in meaning. Both can mean to keep an eye on or to watch carefully.

• iemand in het oog houden
• iemand in de gaten houden

In many situations, they are interchangeable. However, in de gaten houden is often a bit more common in everyday spoken Dutch.

So:

• Houd de kinderen in het oog.
• Hou de kinderen in de gaten.

Both are natural, though the second may sound a little more colloquial.