На мне сегодня белая футболка, а у сестры — синяя.

With clothes, Russian very often uses the pattern на ком? + clothing item:

• На мне белая футболка = literally On me is a white T-shirt
• Natural English: I'm wearing a white T-shirt

So на мне is an idiomatic way to talk about what someone has on.
You could also say:

• Я сегодня в белой футболке = I'm in / wearing a white T-shirt today

That is also natural, just a slightly different structure.

It is the prepositional case after на, because на here means on in the sense of location.

The pronoun я changes like this:

• nominative: я
• genitive: меня
• dative: мне
• prepositional: мне

So the form мне can be both dative and prepositional.
In на мне, it is prepositional.

У сестры uses у + genitive and gives a meaning like:

• my sister has a blue one
• my sister's is blue

It is a very natural way to contrast what one person has with what another person has.

So the sentence is structured a bit like:

• On me today is a white T-shirt, and my sister has a blue one

You could also say на сестре синяя футболка if you want the same on someone structure as the first half. But а у сестры — синяя is a neat, concise contrast.

Also note the case:

• сестра → у сестры = genitive after у

Yes. Russian often leaves out the present-tense linking verb where English would use is / am / are.

So:

• На мне сегодня белая футболка
  literally feels like On me today [is] a white T-shirt

In the second half, even more is omitted because Russian avoids repeating obvious words:

• а у сестры — синяя
  = and my sister's [T-shirt is] blue / and my sister has a blue one

This kind of omission is extremely common and natural in Russian.

Because the noun is already clear from context.

Russian, like English, can omit a repeated noun when an adjective makes the meaning obvious:

• белая футболка, а у сестры — синяя
• English: a white T-shirt, and my sister's is a blue one

So синяя here really means синяя футболка.

This is very common in Russian:

• У меня чёрная машина, а у него — белая.
• I have a black car, and he has a white one.

They agree with футболка, which is:

• feminine
• singular
• nominative

So the adjectives must also be feminine singular nominative:

• белый → белая
• синий → синяя

That is because футболка is the thing being described.

The dash marks an ellipsis: some words are being left out because they are already understood.

In full, the idea is something like:

• На мне сегодня белая футболка, а у сестры [футболка] синяя.

The dash helps show that the second half is shortened and contrastive.
It sounds natural and elegant in Russian.

So the dash is doing something like English intonation in:

• I'm wearing a white T-shirt today, and my sister — a blue one.

Yes, сегодня means today, and Russian word order is flexible.

You could also say:

• Сегодня на мне белая футболка, а у сестры — синяя.
• На мне белая футболка сегодня, а у сестры — синяя.

The original sentence is very natural. Putting сегодня after на мне gives it a smooth, conversational flow.

In general, moving сегодня changes emphasis slightly, not the basic meaning.

Yes. Some natural alternatives are:

• Я сегодня в белой футболке, а сестра — в синей.
• На мне сегодня белая футболка, а на сестре — синяя.

These all mean basically the same thing, but the feel is a little different:

• на мне / на сестре focuses on what is on someone
• я в футболке focuses on what someone is dressed in
• у сестры — синяя has a compact, contrastive my sister's is blue feel

So the original sentence is not the only option, but it is perfectly natural Russian.