I think these links best explain what a core is, but I'll just summarize:
A core is a synergy of two or more Pokémon that get a specific job done, whether it be to Stall Break, cover each others threats in a Meta Game, or take hits and set hazards. Additionally, there are two types of cores: Offensive and Defensive Cores.
Offensive cores tend to cover as many threats as possible, and as effectively as possible. They are generally effective at covering each other's checks and counters. Offensive cores typically consist of Tanks and Sweepers.
Defensive cores tend to have great Type synergy that allows different Pokémon to switch into each other's attacks and typically do more damage through Entry Hazards, Status, and other passive damage than Attacking moves.
Also to note, the best cores typically incorporate both Type synergy and Threat coverage, although they are typically aimed at one or the other. :P
Cores also don't use any Pokémon: they generally use viable options for the job (you wouldn't use a Pikachu When you could use a Zeraora, for example). Sometimes an option is preferred over another, more viable option to cover threats more effectively, but the slightly less viable Pokémon is usually still in the Tier, it's Border Line, or occasionally, the Tier below. (Ubers Viability here) :P
One of team building's hardest challenges is core development, and most (if not all) people don't know it intuitively, but with time, you'll find the cores you are looking for.
Source: Links above, and EXP
Hope I Helped!