The idea is to tax land as a percentage of its value (notably, not its size). This is because land is a necessary good which is exclusionary: because there is a fixed supply of land, it can only be owned at the expense of someone else.

So, let's take the scenario you gave: it depends very much where the mansion and the trailer are. Generally, people build mansions on high-value land and trailers are on low-value land, so the mansion will be taxed more than the trailer (a half acre in LA has a much higher value than a half-acre in rural Nevada where I live). If they are on the exact same lot, though, then yes they would pay the same tax. The idea is that taxing this way incentivizes the best use of land. Imagine that you have identical plots in New York, one is a big apartment complex, and the other a parking lot. With a property tax, you take a huge tax penalty for building, and so the housing complex pays a lot of taxes and the parking lot almost nothing. With a land value tax, however, the parking lot and the housing complex pay the same amount in tax, so it incentivizes the owner of the lot to build a more profitable housing complex there as well, and they take no tax penalty.

Prop 13 in CA is a law that freezes property tax assessments at sale, so you have people who are paying taxes on a home currently worth $2 million as if it were only worth the $150k they bought it for in 1980. This means it is always better just to rent the house out than sell it, since you have a huge tax benefit, making housing almost impossible to buy. Worse, you can pass the frozen assessment on in your will. So it is almost literally creating a quasi-landed gentry class.