Minions are a type of combat method that you summon with Summon weapons called Staves (except for a couple exceptions).
Once summoned, a minion floats/walks around with you passively and attacks enemies of their own accord with no cost or attention needed from you. You can keep fighting or going about your business, using whatever weapons you want. They will not disappear or go away unless you die, leave the game, or desummon them. The damage they deal is Summon damage, and is boosted by Summon Damage stat boosters from Armor or Accessories.
With base stats, you can have 1 minion at a time, but you can increase this Minion Slot number with equipment and temporary buffs. You can mix and match different types of minions, you can just summon a bunch of the same, and there are a few minions that you can spend more than 1 slot on to "power up" instead of having more than one. Abigail, for instance, gets stronger for every slot you spend, but you'll never have more than 1 Abigail.
So how do Whips factor in?
Whips are a hand-weapon that is used actively by the player, dealing Summon damage. In addition to dealing damage directly, they have a system called "Tagging", where hitting an enemy with a whip will temporarily debuff that enemy so that it takes extra damage from Minions. For example, hitting an enemy with a Leather Whip will deal 14 Summon damage, but for the next few seconds, that enemy will take +4 damage from every Minion attack. So if your Minion normally does 5 damage per hit, they will be dealing 9 damage per hit until the tag duration runs out. You can refresh Whip tags by continuing to hit them. On top of that, whipping an enemy will direct the attention of your minions, making them target the most recently whipped enemy. This allows you some control over your otherwise unruly minion targetting.
So in a normal combat scenario, the player (you) would be attacking enemies with whips, while your minions would be running around attacking things, probably the thing you are whipping, and dealing extra damage based on the whip you use.
There are also a different type of minion called Sentries. Sentries are like minions that stay in one spot and attack, like a sort of magical turret. They don't last forever, instead lasting for 10 minutes after you summon them. They use their own capacity counter (Sentry Slots) that doesn't interfere with Minion Slots, but extra Sentry slots are harder to come by, being present on less gear than Minion Slots are. They also deal Summon damage, and also benefit from Whip tagging. You can use Whips, Minions, and Sentries at the same time.