A base core is a pair of pokemon that work well together either offensively, defensively, or both.
EG of defensive
Ferrothorn + Jellicent: One can easily switch between them and cover all of each other's weakness for simplicity's sake. Jellicent has Elec, grass, Dark, and Ghost Weakness. Ferro's Steel/Grass type covers that. Ferro is weak to Fighting and Fire moves. Jellicent resists fire and is immune to fighting
EG of Offensive
Magnezone + Dragon pokemon: Example of offensive, the DragMag core. The only types that resist Dragon is Steel, and Magnezone removes those steel types so Dragons can wreck everything with insanely strong Dragon-moves. As in some cases, they don't have much defensive synergy between them, such as using Hydreigon or Kingdra with it.
EG of both Offensive and Defensive
Rotom-W and Scizor: A Volt-Turn core that shows great defense and offense synergy. Rotom resists Scizor's fire weakness and Scizor resists Rotom's Grass weakness. Offensively, The combination of Volt Switch and U-Turn allows for Scizor to damage and switch into the other. For example, the opponent switches Skarmory out, which resists U-Turn. Scizor U-Turns out and goes into Rotom-W, whose Volt Switch automatically threatens Skarm for KO potential, and with the other moves both pokemon carry, they can play mind games with opponents as to what they're going to use next. While it isn't all that simple and straightforward, it's the best way for you to understand how it works.
Hazards are the moves Stealth Rock, Spikes, and Toxic Spikes. They are very important in competitive battling because a lot of the time, you will be switching Pokemon in and out. Hazards damage pokemon that are switching into the battlefield. These moves are really useful for all playstyles. It makes KOing opponents for Offensive teams much easier. For Defensive teams, their goal is to force the opponents out through defensive strength and let hazards hurt them everytime they switch.
Spikes can be "stacked", or added to increase its damage output. Spikes can be stacked 3 times, the first stack doing 6% for every switch, the second stack doing 12% for every switch, and 25% for every switch. However, it cannot hurt pokemon that are part Flying type or have Levitate or Magic Guard as their abilities.
Stealth Rock can only be stacked once. It does damage to Pokemon based on their weakness to Rock-types. Those who resist Rock 4x e.g. Lucario only get 3% damage. Those who resist Rock 2x e.g. Magnezone only get 6% damage. Those who are neutral to Rock e.g. Blissey get 12% damage. Those who are weak to Rock 2x e.g. Staraptor get 25% damage. Those who are weak to Rock 4x e.g. Moltres get hit with 50% damage, or half of their total HP taken away from their current HP(!). However, this doesn't hurt pokemon with Magic Guard
Toxic Spikes can be stacked twice. It induces a Poison status on opponents. The first stack only normal Poisons the opponent, which means it does 12% to your health every turn. The second stack badly Poisons the opponent, which means that it does 6% the first turn and gradually increases the damage done every turn. However, flying types, steel types, pokemon with levitate, pokemon with immunity, or pokemon with magic guard are not hurt by it. Furthermore, a Poison type can remove Toxic Spikes by switching in.
To get rid of hazards, you need the move Rapid Spin. Rapid Spin removes all hazards upon doing damage, which means Ghost types can stop opponents from removing hazards, which is a tactic called Spin-Blocking.
I hope this helps in some remote fashion