How to remove stains from tile?

Presumably if it was easy it would have been removed in the normal process of cleaning between tenants, unless the landlord doesn't bother with those minor details. I'd be inclined to try a thick paste of baking soda (sodium bicarbonate) and water applied and left overnight, then scrub, rinse, (rinsing is very important before moving on to the next step, if you need to go there) if stain still there move to chlorine laundry bleach.

Can't really tell from a picture if that's stone (a gray, concrete-like stone) or concrete.

I like to use a weak solution of Muriatic acid and water 10% acid. Put it on the stain and rub it in, then rinse at least 3 times with fresh water 3 times. Stronger solutions up to 30% can be used but it will etch the stone / cement tile. Remember to always add acid to water. Fresh air is always a good idea with any chemical and gloves. Muriatic acid can be purchased at most home stores and all pool supply locations. It is a good idea after dry to seal the tile.

If it is stone tile then it is porous and can be very hard to remove some kinds of stains. If it is ceramic porcelain tile then it should be easier but you did say bathroom and tiles used for bathroom floors should be slip-resistant and that texture can make removing stains difficult.

Can you determine if it is an oily or waxy stain? These kinds of stains respond best to non-water based solvents. There are many out there in the paint section of home improvement stores and some of them can also cause staining of porous stone with a oily residue (turpentine for example). Sometimes it takes one solvent to take out or loosen up a stain and other to take out the solvent. Brushing with small brush immune to the solvent you are using and eye protection is helpful.

Or is more like a mineral or rust stain? Sulfamic acid is available in the tile section of home improvement stores and is used for taking haze, effervescence, rust and other mineral staining from tile.

Organic stains like blood, beet juice, etc. can sometimes be successfully removed with enzymatic spot cleaners for carpets, it can take a while. I clean, WHITE (so dyes can't bleed out), terry cloth rag kept wet with the enzymatic cleaner for a while can do surprising things for some stains that seem impossible to remove.

Remember to avoid mixing chemicals that you don't know for sure can be mixed. Always wear gloves that you know are compatible with the chemicals you use. Protect your eyes. You will be sorry if a splash or droplets from a brushes bristles reach your eyes. Most solvents and other products I've mentioned are toxic so keep pets and kids away. Open a window to avoid the build up of poisonous fumes.

For Coffee, tea, or juice: Wash your stains with detergent and hot water, then blot with hydrogen peroxide or diluted bleach.

For Gum, wax, or tar: Put ice cubes in a resealable plastic bag and keep the bag on the material you want to remove from the tile. Once the material is firmly formed, remove it as much as possible with a crafts stick. Remove residual residue with paint thinner.