Co znaczy What does “Ci” mean in every context? ?

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"Ci" can act as a demonstrative pronoun (1), a personal pronoun (2), or a particle (3.)

1) It's the masculine personal plural form of "ten" (this), so in other words it means "these" and is used with masculine nouns in the plural that denote people.

Ci mężczyźni (these men)
Ci chłopcy (these boys)
Ci panowie (these gentlemen)
Ci żołnierze (these soldiers)
Ci robotnicy (these workers)

2) It's the short (unstressed) dative form of "ty" (the stressed form being "tobie.")

Dam ci klucze
(I will give you the keys)

Kupiła ci prezent
(she bought you a gift)

Przyniosłem ci jedzenie
(I brought you food)

Książkę dałem tobie, nie jemu
(I gave the book to you, not to him - in this sentence the logical stress falls on you, so we use the stressed, long form)

3) It's an emphatic particle that can be loosely translated as "what a" or "that's some." It's very often used with "(a) to" followed by "dopiero" ((a) to ci dopiero), and it's safest to stick to this construction if you want to use it.

A to ci dopiero
(well, well, well)

To ci dopiero miasto
(this is quite a city)

To ci była dopiero wycieczka
(that was some excursion)

A to ci historia
(now there's a story for you)

Masz ci los
(oh dear / that's fate for you)

Ten ci umie śpiewać
(he can sing, and how!)