Mi ne memoras vian telefonnumeron.

Because it is the direct object of the verb memoras.

In Esperanto, the thing directly affected by the verb usually takes -n. Here, the thing being remembered or not remembered is your phone number, so telefonnumero becomes telefonnumeron.

• Mi memoras la nomon. = I remember the name.
• Mi ne memoras vian telefonnumeron. = I don’t remember your phone number.

Because via is an adjective-like possessive word, and in Esperanto adjectives agree with the noun they describe.

Since telefonnumeron is accusative and ends in -n, vian must also take -n.

So:

• via telefonnumero = your phone number
• vian telefonnumeron = your phone number, as a direct object

This agreement is very important in Esperanto.

In Esperanto, ne usually goes directly before the word it negates. Here it negates the verb:

• Mi memoras = I remember
• Mi ne memoras = I do not remember

That is the normal and most natural placement.

Because Esperanto uses the present tense -as for this same idea.

The verb memori means to remember, and:

• mi memoras = I remember
• mi ne memoras = I don’t remember

This matches normal English usage quite well. Esperanto does not need an extra helping verb like do.

The basic dictionary form is memori, which is the infinitive and means to remember.

Esperanto verb endings are very regular:

• memori = to remember
• memoras = remember / am remembering
• memoris = remembered
• memoros = will remember
• memoru = remember! / let ... remember
• memorus = would remember

So memoras is just memori with the present-tense ending -as.

Because Esperanto commonly forms compound words by joining roots together.

Here the parts are:

• telefono = telephone
• numero = number

Together they form telefonnumero = telephone number / phone number.

In the sentence, it becomes telefonnumeron because of the accusative -n.

Also, you see a double n in telefonnumeron because the first part contributes an n and then numero begins with n.

Because possessive words like mia, via, lia, ŝia, nia, ilia usually make the meaning definite by themselves.

So Esperanto normally says:

• mia libro = my book
• via telefono = your phone
• vian telefonnumeron = your phone number

Using la with a possessive is usually unnecessary and often unnatural in a basic sentence like this.

Yes, Esperanto word order is fairly flexible because the -n ending marks the object.

The most neutral order is:

• Mi ne memoras vian telefonnumeron.

But other orders are possible for emphasis, such as:

• Vian telefonnumeron mi ne memoras.

That still means I don’t remember your phone number, but it gives extra emphasis to your phone number.

For learners, the standard order subject + ne + verb + object is the best one to use first.

In Esperanto, the verb ending shows the tense, but it does not change for different persons the way it does in some languages. So the subject pronoun is normally stated.

• mi memoras = I remember
• vi memoras = you remember
• li memoras = he remembers

The verb stays memoras each time, so mi is needed to show who is doing the action.

Memori normally means to remember or to have in memory.

If you want to memorize / learn by heart, Esperanto more often uses something like:

• lerni parkere = learn by heart
• enmemorigi = cause to be memorized / commit to memory

So in this sentence, Mi ne memoras vian telefonnumeron, the meaning is clearly I don’t remember your phone number, not I’m not memorizing your phone number.

Yes, but it means something slightly different.

• Mi ne memoras vian telefonnumeron. = I don’t remember your phone number.
• Mi forgesis vian telefonnumeron. = I forgot your phone number.

The first focuses on your present lack of memory. The second says that forgetting happened.

In many situations they may feel similar, but they are not exactly the same.