Me siento débil hoy.

Because the verb here is sentirse, which means to feel in the sense of to feel a certain way.

• me = myself / I
• siento = I feel

So me siento débil literally means I feel myself weak, which in natural English is I feel weak.

If you say just siento, it usually needs an object or means something different, for example:

• Siento frío = I feel cold
• Lo siento = I’m sorry

But for personal states with an adjective, Spanish very often uses sentirse:

• Me siento bien = I feel well
• Me siento cansado = I feel tired

Siento is the first-person singular present tense form of sentir / sentirse.

The infinitive is sentir = to feel.

Its present tense is irregular because it has a stem change:

• yo siento
• tú sientes
• él/ella se siente
• nosotros nos sentimos
• vosotros os sentís
• ellos/ellas se sienten

So siento means I feel.

Spanish often leaves out subject pronouns because the verb ending already shows who the subject is.

• Me siento débil hoy = I feel weak today
• Yo me siento débil hoy = also correct, but yo adds emphasis or contrast

For example:

• Yo me siento débil, pero él se siente bien.
  I feel weak, but he feels fine.

So leaving out yo is completely normal and natural.

This is a very common question.

• sentir often means to feel something or to regret
• sentirse means to feel in a personal condition or state

Examples:

• Siento dolor. = I feel pain.
• Lo siento. = I’m sorry.
• Me siento débil. = I feel weak.

So in your sentence, sentirse is used because the speaker is describing their current state.

Yes, you can, and it is natural. But there is a slight nuance.

• Me siento débil hoy = I feel weak today
  This focuses on how the person feels.
• Estoy débil hoy = I am weak today
  This sounds a bit more like stating a condition or fact.

In many everyday situations, both work.
If you want the closest match to English I feel weak today, me siento débil hoy is an excellent choice.

In the singular, débil stays the same for both masculine and feminine.

• Me siento débil = said by a man
• Me siento débil = said by a woman

In the plural, it changes to débiles:

• Nos sentimos débiles = We feel weak

So unlike adjectives such as cansado / cansada, débil does not change for gender in the singular.

The accent mark shows where the stress goes: DÉ-bil.

Without the accent, debil would follow normal Spanish stress rules and would be pronounced differently, with stress on the last syllable.

So the written accent is necessary to show the correct pronunciation and spelling:

• débil = correct

Yes, hoy can move.

These are all possible:

• Me siento débil hoy.
• Hoy me siento débil.
• Me siento hoy débil. — possible, but less common

The most natural options are usually:

• Me siento débil hoy
• Hoy me siento débil

Putting hoy first can emphasize today a bit more.

A simple approximation for an English speaker is:

meh SYEN-to DEH-bil oy

A few useful notes:

• h in hoy is silent
• siento has ie, pronounced like ye in one syllable: SYEN
• débil is stressed on the first syllable: DEH-bil
• hoy sounds roughly like oy

A more accurate pronunciation is approximately: meh SYEN-to DEH-vil oy

In Spain, the b in débil is usually a soft sound between English b and v.

Here, débil means weak, usually in a physical sense.

So Me siento débil hoy often suggests things like:

• low energy
• lack of strength
• feeling faint
• feeling unwell

Depending on context, it could also be emotional, but most people would first understand it as physical weakness.

Yes, completely natural.

A person in Spain could say:

• Me siento débil hoy.
• Hoy me siento débil.

Both sound normal and idiomatic. There is nothing unusual or non-Spanish-Spain about this sentence.