Это сокращение понятно друзьям, но не всегда понятно преподавательнице.

Because сокращение is a neuter singular noun. The demonstrative этот has to agree with the noun in gender, number, and case.

Its nominative singular forms are:

• этот for masculine
• эта for feminine
• это for neuter

So with сокращение, the correct form is это.

Here понятно is the short-form predicate corresponding to понятный (understandable / clear).

A very common Russian pattern is:

• что-то понятно кому-то = something is understandable to someone

The form changes to match the subject:

• текст понятен — masculine
• идея понятна — feminine
• слово понятно — neuter
• правила понятны — plural

Since сокращение is neuter singular, понятно is the correct form.

A useful contrast:

• понятное сокращение = an understandable abbreviation
  full form, used before a noun
• сокращение понятно = the abbreviation is understandable
  short form, used as the predicate

Because with понятен / понятна / понятно / понятны, the person who finds something understandable is usually put in the dative case.

So Russian asks:

• понятно кому? — understandable to whom?

That gives:

• друзьям — to friends
• преподавательнице — to the female teacher

This is different from the verb понимать:

• Друзья понимают сокращение.
  Here сокращение is the direct object.

But in your sentence, Russian is not using understand as an action verb. It is describing the abbreviation as being understandable or not understandable to certain people.

Because Russian normally omits the verb to be in the present tense.

So:

• Это сокращение понятно друзьям.
  literally looks like This abbreviation understandable to friends

But grammatically it means:

• This abbreviation is understandable to friends.

In the past or future, Russian does use a form of to be:

• Это сокращение было понятно друзьям.
• Это сокращение будет понятно друзьям.

Russian often repeats the predicate in contrasting structures, especially when each half has a different person or condition attached to it.

So:

• понятно друзьям
• но не всегда понятно преподавательнице

The repetition makes the contrast clear and balanced.

If you removed the second понятно, the sentence would sound much more elliptical:

• Это сокращение понятно друзьям, но не всегда преподавательнице.

A native speaker might still understand it in context, but the repeated понятно is the more complete and natural standard version.

It is the dative singular form of преподавательница, meaning female teacher / female instructor / female lecturer, depending on context.

So:

• nominative: преподавательница
• dative: преподавательнице

The sentence is specifically talking about one female teacher.

If it were a male teacher, you would usually have:

• преподавателю

If it were several teachers:

• преподавателям — to teachers
• преподавательницам — to female teachers

Yes. Russian word order is flexible, and changing it changes emphasis more than basic meaning.

The original sentence starts with Это сокращение, so the abbreviation is the topic.

Other possible orders include:

• Друзьям это сокращение понятно, но преподавательнице не всегда понятно.
• Это сокращение друзьям понятно, но преподавательнице не всегда понятно.

These are all possible, but they highlight different parts of the message.

The original version sounds natural because it presents:

1. the thing being discussed — это сокращение
2. who understands it — друзьям
3. the contrast — но не всегда понятно преподавательнице

It modifies the second понятно.

So the idea is:

• not always understandable to the teacher

It does not mean that the teacher never understands it. It means that understanding is inconsistent or depends on context.

You could also place it differently:

• ...но преподавательнице не всегда понятно.

That is also natural. The meaning stays the same, with only a slight shift in emphasis.

Yes, and that would be grammatical, but the focus would be a little different.

Compare:

• Друзья понимают это сокращение.
  Focus: the friends perform the action of understanding.

• Это сокращение понятно друзьям.
  Focus: the abbreviation is understandable to the friends.

Russian often prefers the понятно кому-то structure when talking about clarity, accessibility, or how easy something is to understand, rather than the act of understanding itself.