There are a few different Italian words used to refer to things happening.

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Capitare (to happen casually):

Capita.
It happens.

Capitare, like “to happen,” is the only verb of the four mentioned in this lesson that can be conjugated in all the persons.

Capito spesso da queste parti.
I happen on this place often.

Nonetheless, even capitare is most frequently used in the third person (both singular and plural).

This week’s episode about Marchesi has a good example of the past participle of capitare. It takes the auxiliary essere rather than avere, and is often used with an indirect object pronoun (not reflexive), as in the following example.

Più tardi mi è capitato di lavorare con lui.

Later on I had a chance to work with him.

Caption 14, L'arte della cucina - L'Epoca delle Piccole Rivoluzioni

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With capitare, the happening is generally by chance, or casual in some way.
 

Succedere (to happen, to occur). When something is really taking place, like an accident, or a discussion, we usually use succedere. This is the most common way to say “to happen.”

Che succede?

What's happening?

-Hanno ritrovato il furgone del pesce di Mussa.

-They found Mussa's fish van.

Captions 47-48, Il Commissario Manara - S1EP5 - Il Raggio Verde

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Like capitare, succedere is often used with an indirect object.
 

Ma se mi succede qualcosa e mi mettono in galera?

But if something happens to me and they put me in jail?

Caption 30, La Ladra - EP. 1 - Le cose cambiano

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Accadere (to happen, to occur) is still another way to say “to happen.” It’s a bit more formal, and might be easier to translate with “to occur,” especially since “occur” is also only used in the third person, has a double “c” near the beginning, and ends with an “r” sound. Another easy translation to remember might be “to befall,” since accadere contains the verb cadere (to fall).
 

Ma ormai non accadrà.

But by now it's not going to happen.

No, non succederà.

No, it's not going to happen.

Captions 16-17, Tiromancino - L'essenziale

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Technically, accadere can be used with an indirect object but it’s not very common. Here is an example:

è ispirata a un fatto realmente accaduto a me.

it was inspired by a real event that happened to me.

Caption 3, Volare - La grande storia di Domenico Modugno Ep. 1

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Avvenire (to come about, to occur, to take place) is used in the third person only, and is somewhat formal. The easiest way to remember it is with the phrase “to come about,” since avvenire contains venire (to come). We cannot use avvenire with an indirect object.
 

La raccolta avviene fra novembre e dicembre.

The harvest takes place between November and December.

Caption 8, L'olio extravergine di oliva - Il frantoio

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In a nutshell:
 

succedere: most common, is used in the third person only (except for when it means “to succeed someone,” as in royalty). Auxiliary verb: essere. May be used with an indirect object (something happens to somebody).

capitare: may be conjugated in all persons followed by a preposition or adverb. Translations for this form: to end up, to turn up.
It also functions like succedere, in the third person. Auxiliary verb: essere. May be used with an indirect object (something happens to somebody).

accadere: is conjugated in the third person only. Auxiliary verb: essere. May be used with an indirect object (something happens to somebody), but is not all that common. Easy translation: to befall. Often used like “to occur.”

avvenire: is conjugated in the third person only. Auxiliary verb: essere. May not be used with an indirect object. Easy translation: to come about. Often used like “to occur.”

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These are very basic “rules,” but keep your eyes and ears open to really get the feel for these verbs. And don’t worry. You can get by in most situations with succedere!