У этой розы ещё маленький бутон, а у лилии уже упал один лепесток.

Russian very often expresses possession with the pattern у + genitive + есть / implied there is.

So:

• У этой розы ещё маленький бутон
  literally means something like By this rose, there is still a small bud.

Natural English: This rose still has a small bud.

This is the normal Russian way to say that someone or something has something.

Because the preposition у requires the genitive case.

So:

• эта → этой
• роза → розы
• лилия → лилии

That is why you get:

• у этой розы
• у лилии

Even though English just says this rose and the lily, Russian changes their forms because of у.

In this kind of possessive/existential sentence, the thing that exists is usually in the nominative in an affirmative statement.

So in:

• У этой розы ещё маленький бутон

the noun бутон is the thing being stated to exist, so it stays in the nominative.

A useful comparison:

• У этой розы есть бутон = This rose has a bud
• У этой розы нет бутона = This rose has no bud

After нет, Russian uses the genitive, but in a positive sentence like this one, бутон is nominative.

Because in the present tense, Russian usually omits есть in neutral statements.

So:

• У этой розы ещё маленький бутон

really means:

• У этой розы ещё есть маленький бутон

But in normal modern Russian, есть is often left out when it is not being stressed.

So the first clause is completely normal even without a visible verb.

They create a time contrast:

• ещё = still
• уже = already

So:

• У этой розы ещё маленький бутон
  = This rose still has a small bud
• а у лилии уже упал один лепесток
  = while the lily has already lost one petal / one petal has already fallen from the lily

The sentence contrasts an earlier stage and a later stage.

Here а shows a contrast or comparison between two situations.

It often corresponds to:

• whereas
• while
• sometimes a mild but

So here it is something like:

• The rose still has a small bud, whereas the lily has already lost one petal.

Compare:

• и = simple addition: and
• но = stronger contradiction: but
• а = contrast between two parallel facts

That makes а the best choice here.

Because упал is perfective, from упасть, and it describes a completed event: the petal fell.

• упал = fell / has fallen
• падал = was falling / used to fall / fell repeatedly depending on context

Here the speaker is talking about a single finished result: one petal has already fallen.
So упал is the natural choice.

Because лепесток is the subject of the verb упал.

Subjects are normally in the nominative case, so:

• один лепесток упал

Both words are nominative masculine singular:

• один agrees with лепесток
• лепесток is the thing that fell

If it were genitive, the structure and meaning would be different.

In the past tense, Russian verbs agree with the subject in gender and number.

The subject here is:

• один лепесток

And лепесток is:

• masculine
• singular

So the verb must be:

• упал

If the subject were feminine, you would get упала.
If it were plural, you would get упали.

Russian does not have a direct equivalent of the English present perfect in the same way English does.

So Russian often uses the ordinary past tense for something that English expresses as:

• has fallen
• has happened
• has arrived

Here:

• уже упал

naturally means:

• already fell
• or in better English, has already fallen

The word уже helps show the idea of a completed result relevant now.

Because у + genitive can show the idea that something belongs to a whole and that the whole is affected.

So:

• у лилии упал один лепесток

means something like:

• the lily has had one petal fall off
• one petal has fallen from the lily

This presents the lily as the flower that has lost a part.

You could also hear:

• С лилии упал лепесток

That version focuses more literally on the physical source: from the lily.
But у лилии is very natural when talking about a part of something being lost.

• бутон = bud, an unopened flower
• лепесток = petal, one of the parts of the flower

So the sentence contrasts two stages:

• the rose is still at the bud stage
• the lily is already at the stage where a petal has fallen

That helps create the overall image of different moments in flowering.