I dag lærer vi at bruge “lige så ... som”, når to ting er ens på en bestemt måde.

Because Danish main clauses normally follow the V2 rule: the finite verb comes in second position.

• Vi lærer i dag = the subject comes first.
• I dag lærer vi = the time expression comes first, so the verb still has to stay second, and the subject moves after it.

So I dag lærer vi is very natural Danish and puts extra focus on today.

Here at is the infinitive marker, like English to in to use.

So:

• lærer vi at bruge = we are learning to use

It does not mean the preposition at in English. It is just the marker used before the basic verb form bruge.

It is the standard Danish pattern for an equal comparison, just like English as ... as.

Examples:

• Han er lige så høj som sin bror. = He is as tall as his brother.
• Hun løber lige så hurtigt som Anna. = She runs as fast as Anna.

So in your sentence, it is explaining a structure used when two things are equal in a certain way.

The adjective or adverb goes between så and som.

Pattern:

• lige så + adjective/adverb + som

Examples:

• lige så stor som = as big as
• lige så gammel som = as old as
• lige så hurtigt som = as quickly/fast as

So the ... in lige så ... som is the word that describes the quality being compared.

Yes, in this comparison pattern it is normally written as two words:

• lige så stor som
• lige så dyr som
• lige så hurtigt som

That is the spelling you should learn for this structure.

It can be used with more than just adjectives.

With adjectives:

• Bilen er lige så dyr som den anden. = The car is as expensive as the other one.

With adverbs:

• Han arbejder lige så hurtigt som Maria. = He works as quickly as Maria.

With nouns, Danish usually uses mange or meget:

• Vi har lige så mange bøger som dem. = We have as many books as they do.
• Hun har lige så meget tid som sin søster. = She has as much time as her sister.

So the idea is the same, but with nouns you often need mange or meget in the middle.

Here når means when.

In this sentence it introduces a general situation:

• når to ting er ens på en bestemt måde = when two things are alike in a certain way

This is different from:

• hvis = if
• da = when (for a specific past event)

So når is the right choice because the sentence is talking about a general rule, not one single past event.

This is a very common question.

ens means alike, equal, or the same in appearance/quality.

• De to tasker er ens. = The two bags are alike.

samme means the very same one.

• Det er den samme taske. = It is the same bag.

So in your sentence, to ting er ens means two things are alike / equal in a certain way, not that they are literally one and the same object.

Because to ting simply means two things in a general, indefinite way.

• to ting = two things
• de to ting = the two things / those two specific things

In your sentence, the meaning is general: whenever two things are equal in some way. It is not talking about two specific things already known to the listener.

på en bestemt måde means in a certain way or in a specific way.

This is an idiomatic Danish expression:

• på en måde = in a way
• på en bestemt måde = in a certain/specific way

The preposition på is just the normal one used with måde in this kind of phrase. English uses in, but Danish uses på.

Yes:

• Min computer er lige så hurtig som din. = My computer is as fast as yours.
• Sofie er lige så gammel som Emil. = Sofie is as old as Emil.
• Han synger lige så godt som sin lærer. = He sings as well as his teacher.
• Vi har lige så mange elever som naboklassen. = We have as many students as the neighboring class.
• De bruger lige så meget tid som os. = They use as much time as we do.

A very useful pattern to remember is:

X er lige så + adjective + som Y

For example:

Huset er lige så stort som skolen. = The house is as big as the school.