First, you need to create Ink items via Alchemy crafting. If your mansion has the relevant upgrades then the recipes should already be visible. Note that these recipes involve rare resources acquired by adventuring (such as Troll's Blood) as well as a vendor item purchased with gold (Ink Base). If you just want to experiment then you can use the Cheat menu to add Ink items to your inventory.
Applying a tattoo consumes 1 unit of the associated Ink. Fully tattooing a single character will require 8 units, but you'll need to mix-and-match the different Inks (because each tattoo is applicable only to certain body parts).
To apply a tattoo, open the Gear menu. There's a special "needle" button in the lower-right corner. The button is always visible (even when you have 0 Ink) so you may have accidentally clicked it in the past. Click that button to switch between "gear" mode and "tattoo" mode.
When you're in tattoo mode, the normal item list (potions, swords, etc) will be replaced by a list of Ink items. You must select an Ink and then click on a body part to apply the tattoo (you'll be prompted to confirm Yes/No). After you've applied a tattoo you can hover the mouse pointer over that slot; the game will display a tooltip showing the tattoo's effects.
You don't need to fill all of the tattoo slots. It's often useful to apply several copies of the same tattoo in different slots (e.g. lots of Mana regen for dedicated spellcasters) but there isn't a "set bonus" so you don't need to worry much about min-maxing.
Although they use the equipment menu, Tattoos cannot be "unequipped". 1 unit of Ink will be consumed when applying a tattoo and there's no way to retrieve it - if you make a mistake then you'll need to Craft more Ink.
You CAN overwrite a tattoo in the future by applying a different Ink to an already-tattooed slot. You CANNOT remove a tattoo entirely (i.e. replace it with nothing) ... but there's little reason to remove tattoos because their in-game effects are entirely positive.