Measurements needed to determine correct tire

The most important measurement is the bead seat diameter. Take a tape measure, and measure the distance from the ground where the wheel is standing to the top of the hook at the topmost part of the wheel. This measurement is not the bead seat diameter but rather the top-of-the-hook diameter of the wheel.

Then, next, take a caliper that can measure the depth of holes. Measure the distance from the top of the hook to the bottom of the bead seat. The bead seat is the seat below the hook, not the "valley" at the middle of the rim but rather the seat at the side of the rim, where the tire bead sits.

Calculate BSD = top_of_the_hook_diameter - 2 * bead_seat_depth

For example, let's consider my fatbike rims. If top_of_the_hook_diameter is 572mm and bead_seat_depth is 7mm, then you get 572 mm - 2 * 7 mm = 558 mm. This would mean the rim bead seat diameter is ISO 559mm, because that's the closest match of the commonly used bead seat diameters. Here's a comprehensive list and some instructions for measuring.

The second measurement is inner rim width. I won't use the fatbike rims as an example because they have a completely ridiculous width. Let's measure the rim width of DT Swiss TK 540, from inside of one hook to inside of another hook: the measurement says 18 mm. However, this is a rim that has hooks at the sides, meaning the width below the hooks is more than at the hooks. Thus, this rim would be called "19C" (C = crotchet-type, i.e. has hooks).

Put the two numbers together. The TK540 rims are not 559mm but 622mm bead seat diameter so I'll use those. The rim size is 622-19C.

The tire bead seat diameter must match the first number, i.e. be 622mm. You have zero flexibility in this: the tire and rim must absolutely match. The tire width should be applicable for 19C rims (although you can possibly get away with slightly exceeding the limits). See tire and rim width compatibility. The 622-19C rims take from 622-28 to 622-44 tires.