Since the wheel was invented, carts have been used for transporting goods. But carts and similar vehicles come in all shapes and sizes and are used for so many different things, so it's natural for there to be variants depending on the size and function.
The basic noun in Italian is carro. Its definition in Italian is this:
Veicolo a trazione animale o meccanica, costituito da un piano sostenuto da due o quattro ruote, usato per il trasporto di materiali e merci.
Vehicle that is mechanical or pulled by animals, constisting of a flat bed supported by two or four wheels, used to transport materials and goods.
The noun carro by itself generally indicates a rather large-sized cart. But what it's used for is usually in the form of a second noun used as an adjective, an adjective, or as an alteration of the word carro.
Carro is used together with a modifier describing its function:
Ho capito la situazione, ma non... chiama il carro attrezzi, il soccorso stradale, qualcosa.
I understand the situation, but I can't... call the tow truck, the roadside assistance, something.
Captions 38-39, Provaci ancora prof! S2E5 Vita da cani - Part 4
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Essentially, you are loading a car on a big "cart" equipped to do that. It's attrezzato (equipped) with gli attrezzi (equipment).
In a parade or procession, called una sfilata, un corteo, un corteo storico, or una processione, the floats are called i carri.
Come vedi c'è il carro di Santa Rosalia e al posto della Santa... -E lui lì sopra.
As you can see, there's the Saint Rosalia [patron saint of Palermo] float and in place of the Saint... -And him on top of it.
Captions 25-26, Dottor Pitrè e le sue storie - Part 2
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For a funeral procession, as well, the noun carro is used. Even today, with a motorized hearse, the same term is employed.
Sarà mica un carro funebre?
It wouldn't be a hearse, would it?
Caption 64, Provaci ancora prof! S1E2 - Un amore pericoloso - Part 12
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Literally, "the armed cart."
Me lo dici che premio è? Un carro armato vero, nuovo nuovo.
Will you tell me what the prize is? A real tank, brand new.
Captions 20-21, Trailer La vita è bella - Roberto Benigni
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We can use diminutive suffixes to indicate a smaller carro.
È su un carretto trainato forse da una pecora, comunque da un ovino ed è un momento felice, anche se per molti esperti, si tratta di un viaggio simbolico verso l'aldilà, il regno dei morti.
He is on a cart drawn by perhaps a sheep, in any case by an ovine and it is a happy moment, even if for many experts, it involves a symbolic journey to the afterlife, the kingdom of the dead.
Captions 53-55, Meraviglie EP. 2 - Part 6
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The constellations we call "the Big Dipper" and "the Little Dipper" are called respectively il Grande carro and il Piccolo carro in Italian.
There are small carts, too, for transporting things by hand.
Un carrello can be the kind you use at the supermarket, but it can be any kind of cart, trolley, or dolly for transporting relatively small items.
Una rana in servizio non dovrebbe spingere il carrello delle clienti.
A frog on duty shouldn't push the customers' carts.
Captions 65-66, Un medico in famiglia Stagione 3 S3EP4 Lo stagno del ranocchio - Part 6
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Be'? Che state a fa' co' 'ste carriole?
Well? What are you doing with these wheelbarrows?
Caption 2, Spot pubblicitario TIM Tribù
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A wheelbarrow with only 2 wheels might be called una carretta, but this word might also refer to a wreck of a car, a jalopy.
Signore? Sono qua, nella carrozza. Dico a Voi.
Sir? I am here, in the carriage. I'm talking to you.
Captions 32-33, Caravaggio EP1 - Part 9
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But the car of a train might also be called un vagone, an Italianized version of "wagon."
Ferma solo a Guidonia, Firenze, Genova e c'è un vagone ristorante per bambini belli che mangiano tutta la pappa.
It stops only in Guidonia, Florence, Genoa, and there is a dining car for nice children who eat all their mush.
Captions 16-18, Un medico in famiglia Stagione 1 EP3 Un cugino in fuga - Part 2
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La carrozzella in this day and age usually refers to a wheelchair, but once, it referred to a buggy.
Qua, tolto il figlio con la madre in carrozzella non s'è visto nessuno.
Here, apart from the son with his mother in a wheelchair, I didn't see anyone.
Caption 15, Imma Tataranni Sostituto procuratore S1 EP3 I giardini della memoria - Part 15
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Una carrozzina may refer to a baby carriage, but some people use it to mean "wheelchair." It's important to consider the context!
La carrozzeria is the chassis of a car, as well as the place where a chassis is repaired.
I fanalini della macchina rotti, le ruote bucate. -Fanalini della macchina rotti. -Per non parlare della carrozzeria.
The taillights of my car broken, the tires slashed. -Taillights broken. -Not to mention the body.
Captions 42-43, Provaci ancora prof! S2E5 Vita da cani - Part 20
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The person who repairs the chassis of a vehicle is called il carrozziere.
Va bene, allora parlo un attimo col carrozziere.
All right. So I'll have a quick word with the body shop guy.
Caption 16, Provaci ancora prof! S1E3 - Una piccola bestia ferita - Part 14
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There are undoubtedly more words stemming from il carro, and feel free to let us know, so we can add them to this lesson!