Someone help me out with the use of "ĉe." It seems to be one of those words that can mean a lot of different things.

When I look it up in the dictionary I get "at, beside, by, with." Furthermore, when reading Nakamura, we encounter the phrase, "Filipo sentas fortan premon ĉe la brusto" when Lernu itself translates as, "Phillip feels a strong pressure on his chest," so I guess "on" then is an additional possible translation.

"At" and "beside" I see could be related. But "by" seems totally different." And I don't see how the same word could also mean "with."

Could someone provide the various meanings of "ĉe" with some examples?

Yes, ĉe is indeed one of those words that can mean a lot of different things. PIV gives no less than 16 definitions for it. Rather than trying to list all possible ways it can be translated to English, I suggest to you to try get the general idea of ĉe. According to the grammar section here in Lernu it denotes:

❞A very close position (touching or not touching). The meaning of ĉe varies according to the context. Often ĉe simple denotes a general location without a precise indication as to its position.❞

or

❞Extreme temporal proximity.❞

Lernu gives respective examples
 

• Ĉe la angulo de la strato ŝi haltis.
  
• Ĉe la komenco de la tagiĝo la arbaro aperis bele kovrita de prujno.

Noteworthy with the spatial definition is that ĉe is used in contrast to "being in", e.g. en la ŝranko, in the closet but ĉe la ŝranko, at/by the closet. Therefore you say li iris ĉe la dentiston, he went to the dentist because he is not going to go inside the dentist (and here of course with accusative of motion).

For further examples see PIV.

Thanks Metsis, that's helpful.

It looks like ĉe is one of those where you probably need to look up a number of examples and just have its used drilled into your brain via repeated exposure.

From the Lernu article you referenced it sounds like it reference both literal physical proximity and, I'm not sure what to call it, time proximity? (e.g. Ĉe la komenco de la tagiĝo la arbaro aperis bele kovrita de prujno.)