The noun pazienza certainly does look a lot like "patience."  And sometimes the two words do mean the same thing, especially when the article is present.

Mi scusi, signorina,

Excuse me, Miss,

però suo cugino, ogni tanto,

but your cousin, every now and then,

mi fa perdere la pazienza.

makes me lose my patience.

Captions 10-11, Sei mai stata sulla luna? - film

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Something to keep in mind: In English, we use a possessive pronoun: my patience. Italians do it differently. They use a definite article la, but the possession happens with an indirect object pronoun. "It makes me lose the patience."

Adjective form paziente

Although the adjective paziente (patient) does exist in Italian, Italians often opt for the noun form.

Ma no, è che ci vuole soltanto un po' di pazienza.

No, it's that you just need to be a bit patient.

Dai fiducia all'allievo e nel momento giusto lo lasci andare. -Sì.

Give the student some confidence, and at the right moment, let him go. -Yes.

Captions 23-24, Sposami - EP 2

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And let's not forget that, similar to English, il or la paziente can also be a noun meaning "the patient." It can have a feminine or masculine article, depending on the gender of the patient.

A me risulta invece che vi conoscesse [sic: conosceste] da prima,

Instead, it is my understanding that you knew each other before that,

e che Lei fosse stata anche sua paziente.

and that you had also been his patient.

Captions 27-28, Il Commissario Manara - S1EP9 - Morte in paradiso

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2 expressions with pazienza

Abbia pazienza/abbi pazienza

It's common in Italy to ask someone to "have patience" but it isn't necessarily patience they are asking for. 

They use the imperative for this, and are asking for your understanding, tolerance, or to bear with them. It can be used with different tones, including sarcasm.

In the following example, Orazio is upset with his wife who barged in on a meeting, and had to apologize to his clients he had to ask to leave. So saying abbia pazienza can be a way of apologizing for an inconvenience. In this case, he also said scusi (excuse me [formal], sorry), but he could have just said abbia pazienza in the way of apologizing.

Scusi, sa, eh, abbia pazienza.

Excuse me, you know, eh, bear with me.

Caption 32, Un Figlio a tutti i costi - film

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Actually, Orazio is also quite annoyed with his client, who wants to get out of paying taxes for reasons not exactly on the up and up. So in this case, and often, especially when the formal version is being used, abbia pazienza, uttered with an exasperated or annoyed tone, is an "excuse me" that's a bit indignant. It's almost a way of saying you are the one losing your patience.

1) How would you say this if you were on familiar terms with other person?

But the expression is also used, for example, when you have an appointment but they make you wait. Someone might say, abbia pazienza as a way of saying, "Sorry we are making you wait." Or if your doctor or lawyer has to answer a call while you are talking to him or her:

Abbia pazienza, devo prendere questa chiamata. (Sorry, I have to take this call).

If someone really does want you to be patient, they might say, Solo un attimo di pazienza. 

Signore, solo un attimo di pazienza,

Ladies, just a moment of patience.

adesso vi facciamo qualche domanda.

Now we're going to ask you some questions.

Captions 67-68, Il Commissario Manara - S1EP11 - Beato tra le donne

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Pazienza all by itself

The second example of an expression is one of those wonderful one-word expressions that say plenty. You will want this in your toolbox, for sure. It's often coupled with a va' be' (short for va bene [all right or OK]), but doesn't need to be.

Mi dispiace. Sabato arrivano quelli della filiale dal Sud America

I'm sorry. Saturday, the people from the South America branch are coming

e purtroppo ho una riunione con loro.

and, unfortunately, I have a meeting with them.

Ho capito.

I understand.

Va' be', pazienza. -Mi dispiace. -Ingegnere?

Oh well, too bad. -I'm sorry. -Sir?

Captions 41-44, Adriano Olivetti - La forza di un sogno Ep.2

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What are some good occasions for saying pazienza as a one-word expression?

You are at a shop and ask for an item you can't find on the shelves. You ask the clerk:

Non trovo la polenta istatanea (I can't find the instant polenta).

Ah, mi dispiace, è terminata (Oh, I'm sorry, we're out of it).

Ah, pazienza. Farò senza (Oh, no big deal. I'll do without it).

Some other ways to translate pazienza in English:

So be it.

Oh well.

Too bad.

Nothing to do about it.

It is what it is. 

Some synonyms for pazienza in Italian:

Non importa (it doesn't matter)

Non fa niente (it doesn't matter)

Fa niente (it doesn't matter)

È lo stesso (it's all the same)

Perhaps as you go about your day, there will be situations in which pazienza could be a comment you make as a reaction to something that didn't go as you wished. You wanted a dash of milk in your coffee, but you're out of it. Pazienza, lo prenderò senza latte. You wanted to watch the news, but you forgot. Pazienza!

Extra Credit

1) Scusa, sai, eh, abbi pazienza.