Incontro is a noun that means, not surprisingly, "encounter," "meeting," "get-together," or "rendezvous." In English, we tend to save the noun "encounter" for special or particular meetings. In Italian, it gets used more often.

Conoscendolo, penso che sia più probabile

Knowing him, I think it's more likely

che si sia fermato qui per un incontro amoroso.

that he stayed here for a hot date.

Captions 46-47, Il Commissario Manara - S1EP3 - Rapsodia in Blu

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The verb form incontrare means "to meet," "to encounter." It often means to bump into someone by chance.

Anna! Ti, ti ricordi quei due signori che abbiamo incontrato prima?

Anna! Do you remember those two gentlemen we met earlier?

Captions 1-2, Anna e Marika - Hostaria Antica Roma

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BANNER PLACEHOLDER

Keep in mind that the first person singular of the verb incontrare is the same as the noun form, incontro.

Quando vado al mare, incontro tanti stranieri.
When I go to the beach,  I meet up with lots of foreigners.

There is a third form which looks exactly like the noun incontro, but is a preposition, and is used together with a second preposition, a (to, at): incontro a (towards).

It’s used in the very common phrase:

Ti vengo incontro.
I’ll come towards you.
I’ll meet you halfway.

This expression also is used when negotiating:

Mi è venuto incontro sul prezzo
He met me halfway on the price.

We say "halfway" but it may be more or less than half, so we could also say "part way." It can mean making a concession, giving a discount, or lowering a price.

Remembering that contro means "against" will help you understand the following example. It’s another figurative use of incontro, and the verb andare (to go) is used: andare incontro (to face, to encounter, to be up against).

Era medico anche lui.

He was a doctor, too.

Si figuri se non sapeva a che cosa sarebbe andato incontro.

Can't imagine he didn't know what he was up against.

Captions 55-56, Il Commissario Manara - S1EP1 - Un delitto perfetto

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Grammatically speaking:
To use incontro a as a preposition, we need a subject, a verb—usually venire (to come) or andare (to go), and an indirect object. If the object is a person or a noun, we use incontro a:

Vado incontro a Maria.
I’m going to walk towards Maria. / I’ll meet up with Maria halfway.

Va incontro alla morte.
He’ll be facing death. / He’s going towards his death.

If we use incontro a with an indirect object pronoun, the preposition a is already included in the object pronoun if the the pronoun is at the beginning of the phrase. If it’s at the end, it needs a preposition:

Ti vengo incontro.
Vengo incontro a te.
I’ll meet you halfway.

Ci vengono incontro.
Vengono incontro a noi.
They’ll meet us halfway.

Le vado incontro.
Vado incontro a lei.
I’ll meet her halfway.

BANNER PLACEHOLDER

For more examples of the expression andare incontro a see this lesson. 

Just for fun:

Ogni giorno vado incontro a delle situazioni diverse. Ieri ho incontrato un vecchio amico, e volevamo programmare un altro incontro. Non potevo immaginare a che cosa si andava incontro, perché per trovare una data, abbiamo incontrato degli ostacoli non indifferenti. In realtà nessuno dei due aveva tempo per andare a casa dell’altro. Infine, ci siamo venuti incontro. Ci vedremo in città, vicino a dove lavora lui, e mi verrà incontro a piedi per farmi strada.

Every day I’m up against different situations. Yesterday I ran into an old friend, and we wanted to schedule another get-together. I couldn’t have imagined what we were up against, because in trying to find a date for it, we ran into significant stumbling blocks. The fact of the matter is that neither of us had time to go to the other’s house, so we met each other halfway on it. We’ll meet in the city, near his office, and he’ll come and meet me part way on foot to show me the way.