Solitudo silvae aviae grata est.

Here is the most likely analysis:

• solitudo — nominative singular, feminine; the subject. It means solitude.
• silvae — genitive singular, feminine; it modifies solitudo and means of the forest.
• aviae — dative singular, feminine; it means to/for the grandmother.
• grata — nominative singular, feminine; a predicate adjective meaning pleasing, dear, welcome.
• est — is, 3rd person singular present of sum.

So the structure is basically:

The solitude of the forest is pleasing to the grandmother.

Because solitudo is in the nominative case, which is the normal case for the subject.

aviae is not nominative here; it is best understood as dative singular, because grata est often means is pleasing to someone, and that someone goes in the dative.

This is one place where Latin works differently from English. English often says:

• The grandmother likes the solitude of the forest.

Latin here says:

• The solitude of the forest is pleasing to the grandmother.

So in Latin, the thing liked is the subject.

Because it expresses possession or association: the solitude of the forest.

The genitive often answers the question of what? or whose?

So:

• solitudo silvae = the solitude of the forest

This is a very common Latin pattern:
noun + genitive

Examples of the same kind of structure would be:

• puella villae = the girl of the house would be wrong, because that does not express a natural relation
• but
• porta villae = the gate of the house
• amor patriae = love of country

Because gratus, grata, gratum regularly takes the dative of the person to whom something is pleasing, welcome, or dear.

So:

• mihi gratus est = he is pleasing to me / welcome to me
• puellae grata est = she is pleasing to the girl
• aviae grata est = it is pleasing to the grandmother

In this sentence, aviae is the person who experiences the feeling.

Because grata must agree with solitudo.

solitudo is:

• feminine
• singular
• nominative

So the adjective must also be:

• feminine
• singular
• nominative

That gives grata.

This is called agreement: adjectives agree with the nouns they describe or complete in gender, number, and case.

Not here.

A learner might first think of English grateful, but in this sentence grata est means is pleasing, is welcome, or is dear.

That is because of the construction:

• something + gratus/grata + dative person

So:

• Solitudo ... aviae grata est
  = The solitude ... is pleasing to the grandmother

If you translated it as is grateful, the sense would be wrong.

That is a very good question, because -ae is an ambiguous ending in the first declension.

A form ending in -ae can be:

• genitive singular
• dative singular
• nominative plural
• vocative plural

So you have to use context and syntax.

Here:

• silvae works well as genitive singular with solitudo:
  the solitude of the forest
• aviae works well as dative singular with grata est:
  pleasing to the grandmother

So the grammar of the whole sentence tells you how to read the endings.

Yes. Latin word order is much freer than English word order because the endings show the grammatical relationships.

So these could express the same basic meaning:

• Solitudo silvae aviae grata est.
• Aviae solitudo silvae grata est.
• Grata est aviae solitudo silvae.

The exact word order can change the emphasis, but the endings still show what goes with what.

In the given sentence, putting est at the end is very normal Latin style.

Because Latin has no definite or indefinite article.

So Latin does not normally have separate words for:

• the
• a
• an

Whether you translate a noun as the forest, a forest, or just forest depends on the context.

So:

• silvae can mean of the forest or of a forest
• solitudo can mean the solitude or a solitude, depending on context

In most normal translations of this sentence, English uses the.

The dictionary forms are:

• solitudo, solitudinis f. — solitude
• silva, silvae f. — forest, wood
• avia, aviae f. — grandmother
• gratus, grata, gratum — pleasing, welcome, dear
• sum, esse — to be

A learner often needs to know these forms because Latin sentences usually contain inflected forms, not the dictionary headwords.

A very literal unpacking is:

• solitudo = the solitude
• silvae = of the forest
• aviae = to the grandmother
• grata est = is pleasing

So, word by word:

The solitude of the forest is pleasing to the grandmother.

That literal structure is often the easiest way to see the Latin grammar, even if a more natural English version might be:

The grandmother likes the solitude of the forest.