Serva mappam de mensa tollit.

Because serva is in the nominative singular, the case normally used for the subject.

• serva = nominative singular, the female slave / maid / servant
• mappam = accusative singular, so it is the direct object
• tollit = he/she/it lifts/takes up/removes

So serva is the one doing the action.

A small extra note: by itself, serva! could also be the imperative of servare (save! / preserve!), but in this sentence the grammar clearly shows it is a noun.

Because mappam is the direct object of tollit.

In Latin, a direct object usually goes in the accusative case.
The noun mappa is a first-declension noun, so:

• nominative singular: mappa
• accusative singular: mappam

English usually does not change the form of a noun for this job, but Latin does. So mappam tells you what the servant is picking up.

Because de is a preposition that takes the ablative case, and mensa is the ablative singular form.

Here:

• de = from / off / down from
• mensa = table in the ablative singular

So de mensa means from the table or more naturally off the table.

This is a very common Latin pattern:
de + ablative

Because the verb and the context make that clear.

The verb tollit means lifts, takes up, removes, or takes away. With that kind of verb, de + ablative naturally means movement down from or off of something.

So in this sentence:

• mappam de mensa tollit = she lifts/removes the napkin from the table

If the sentence were about speaking or writing, then de would often mean about / concerning instead.

Tollit is the 3rd person singular present active indicative of tollere.

That means it translates as:

• he lifts
• she lifts
• it lifts

Depending on context, tollere can mean several related things:

• lift
• pick up
• remove
• take away

So here tollit could be understood as picks up or removes.

Because Latin word order is much freer than English word order.

English depends heavily on order:

• The servant takes the napkin

Latin depends more on case endings, so the roles of the words are already marked:

• serva = subject
• mappam = object
• mensa after de = object of the preposition

That means Latin can move words around for style or emphasis. Putting the verb at the end is very common in Latin prose.

So Serva mappam de mensa tollit is a very normal Latin order.

Yes, often they could.

For example, these would still basically mean the same thing:

• Serva mappam de mensa tollit
• Mappam serva de mensa tollit
• De mensa mappam serva tollit

The endings still show who is doing the action and what is being acted on.

However, the order can change the emphasis:

• putting mappam early may emphasize the napkin
• putting de mensa early may emphasize where it is being taken from
• putting tollit at the end can give a neat finish to the sentence

So the meaning stays mostly the same, but the focus may shift.

Because Classical Latin normally has no articles.

So:

• serva can mean a servant or the servant
• mappam can mean a napkin or the napkin
• mensa can mean a table or the table

The context tells you which is meant.

That is why the same Latin sentence can be translated in slightly different ways, such as:

• The servant picks up the napkin from the table
• A maid takes a napkin off the table

They are:

• serva — nominative singular
  used for the subject

• mappam — accusative singular
  used for the direct object

• mensa — ablative singular
  used after the preposition de

So the sentence is a nice example of three very important Latin patterns:

• nominative for the doer
• accusative for the thing directly affected
• ablative after certain prepositions such as de