The verb sentire can be quite confusing due to its wide variety of meanings.

1. sentire literally means ‘to sense’, and is used to describe four of the five senses: l’udito (hearing), il tatto (touch), il gusto (taste), and l’olfatto (smell). When used to describe perceptions its definition is contextual, as can be seen in the examples below:
ho sentito un rumore = I heard a noise;
senti com’è morbida questa stoffa = feel how soft this fabric is;
fammi sentire quel formaggio = let me taste that cheese;
si sente il profumo del gelsomino entrare dalla finestra = you can smell the scent of the jasmine coming in through the window.

“si sente il profumo del gelsomino entrare dalla finestra” Photo: Geoff Chamberlain 2013©

However, hearing is the sense most commonly described by the verb sentire even though udire (to hear) is more technically correct:
pronto, mi senti? = hello, can you hear me?
non sento molto bene = I can’t hear very well
sentiamo cosa ha da dire = let’s hear what he has got to say
ci sentiamo domani = we’ll get in touch tomorrow (literally: we’ll hear from each other tomorrow)

sentire, with the meaning of ‘to feel’,  is also used to describe physical sensations:
sento freddo = I’m feeling cold
non senti caldo con quel maglione? = don’t you feel hot with that jumper on?
sento un languorino allo stomaco = I feel a bit peckish (literally: I feel a little emptiness to the stomach)
sentiamo la stanchezza del viaggio = we’re feeling tired from the journey (literally: we feel the tiredness from the journey)

Then there’s sentirsi, reflexive form of sentire:

2. sentirsi also describes how we feel. However, whereas sentire puts the emphasis on an external stimulus and is normally constructed with a direct object (sento la stanchezza = I feel the tiredness), sentirsi describes the action of listening to your own body or emotions and tends to be built with adjectives or adverbs (mi sento stanca = I feel tired). Here are some more examples of its use:
stamattina mi sento bene = this morning I feel well
Lucia non è venuta perché si sentiva stanca = Lucia didn’t come because she was feeling tired
Ciao, come ti senti oggi? = Hi, how are you feeling today?
mi sento proprio un idiota = I feel like a total idiot

Finally we have sentirsela and sentirselo, which belong to a group of verbs called ‘verbi pronominali’ in which one or more pronouns attach themselves to the verb giving it a new meaning:

3. sentirsela is a colloquial form of sentirsi meaning ‘to feel like it’, ‘to feel up to it’. Here, the pronoun ‘la’ becomes part of the verb and doesn’t have any real meaning:
te la senti di andare a fare due passi? = do you feel like going for a stroll?
non me la sono sentita di parlargli = I didn’t feel like talking to him
Giorgio non se la sente di venire = Giorgio doesn’t feel up to coming

4. sentirselo, constructed with ‘lo’ instead of ‘la’, means ‘to have a premonition’:
me lo sentivo che avrei vinto = I knew I would win (literally: I had the premonition I would win)
Gianna se lo sentiva che non avrebbe passato l’esame = Gianna knew she wouldn’t pass the test
andrà tutto bene, me lo sento! = everything will be fine, I know it!