Puella vaccam, quam mater cotidie mulget, libenter tangit.

Because vaccam is the direct object of tangit.

The verb tangere means to touch, so the person or thing being touched goes into the accusative case.
Here:

• puella = the girl, nominative, the subject
• vaccam = the cow, accusative, the object

So Puella vaccam tangit means The girl touches the cow.

Because puella is in the nominative case, which is the case normally used for the subject of the sentence.

Also, the verb tangit is third person singular, so it matches a singular subject: puella.

So the main clause is:

• Puella ... tangit = The girl ... touches

Quam is a relative pronoun, meaning whom, which, or that, depending on the English translation.

It refers back to vaccam.

So:

• vaccam, quam mater cotidie mulget
  means
• the cow, which the mother milks every day

In English we often use that or which. Latin uses the relative pronoun very regularly.

This is a very common question.

A Latin relative pronoun gets:

• gender and number from the noun it refers to
• case from its job inside its own clause

Here, quam refers to vaccam, so it is:

• feminine because vacca is feminine
• singular because vaccam is singular

But inside the relative clause, quam is the object of mulget:

• mater cotidie mulget = the mother milks ...
• what does the mother milk? the cow

So the pronoun must be accusative feminine singular: quam.

If it were the subject of the relative clause, then you would expect quae.

Mater is nominative singular.

In the relative clause, mater is the subject of mulget:

• mater mulget = the mother milks

So even though vaccam is the object in the main clause and quam is the object in the relative clause, mater is the one doing the milking, so it is nominative.

Mulget is:

• third person singular
• present tense
• active voice
• from mulgere = to milk

So mater cotidie mulget means the mother milks every day.

Because it is singular, it matches mater.

Cotidie means daily or every day.

It is an adverb, so it modifies the verb mulget.

Thus:

• mater cotidie mulget = the mother milks every day

Adverbs in Latin are often quite movable, so you may see them in different positions.

Libenter means gladly, willingly, or with pleasure.

It is an adverb modifying tangit.

So:

• Puella vaccam libenter tangit = The girl gladly touches the cow

It tells you how she touches the cow.

Latin word order is much more flexible than English word order because the endings show grammatical function.

So Latin can put the relative clause right after the noun it describes:

• vaccam, quam mater cotidie mulget, ...

This is very natural in Latin. It keeps the description close to vaccam.

In English we do something similar:

• The girl touches the cow, which the mother milks every day

So although the sentence may look interrupted, the structure is normal.

Yes. Latin often allows several word orders without changing the basic meaning.

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

• Puella libenter vaccam, quam mater cotidie mulget, tangit.
• Vaccam, quam mater cotidie mulget, puella libenter tangit.

The case endings still tell you who is doing what.

However, word order can affect emphasis. The given sentence begins with puella, so the girl is somewhat foregrounded.

Classical Latin does not have articles like English the and a/an.

So:

• puella can mean girl, a girl, or the girl
• vaccam can mean a cow or the cow
• mater can mean a mother or the mother

You figure out which is best from context.

A helpful way is to separate the main clause from the relative clause.

Main clause:

• Puella vaccam libenter tangit.
• The girl gladly touches the cow.

Relative clause:

• quam mater cotidie mulget
• which the mother milks every day

Then you insert the relative clause next to the noun it describes:

• Puella vaccam, quam mater cotidie mulget, libenter tangit.

This is often the easiest way to read Latin sentences with embedded clauses.

Not in the same way that endings matter.

The really important grammatical signals in Latin are the word endings, not the commas. Punctuation in printed Latin is mostly an editorial aid for readers.

Here, the commas help you see that quam mater cotidie mulget is a parenthetical relative clause describing vaccam.

So the commas are helpful, but the grammar would still be understandable from the forms:

• vaccam = object in main clause
• quam = object in relative clause
• mater = subject in relative clause
• tangit = main verb
• mulget = verb of relative clause