Caleb's piece covers the ground well, but I want to look specifically at the economic welfare effects of rent control, and who gains and who loses from controlled rents. Remember that governments (local or central) typically enact rent controls in order to help low-income tenants who may be struggling to pay high rents (often in inner-city neighbourhoods).
Consider the market in the diagram below. If the market is at equilibrium, the market rent is R0, and the quantity of housing rented is Q0. Now say instead that there is a binding rent control at R1 (below the market rent). The lower rent makes renting more attractive relative to owning your own home. Some people would find it cheaper or more convenient to be a renter at this lower rent, so the quantity of rental housing demanded increases (to QD). However, the lower rents make rental housing a less attractive investment for landlords. Perhaps they convert that rental housing into commercial rentals instead (e.g. offices) or maybe they choose to live there themselves (the opportunity cost of living in the house is now lower). Either way, the quantity of rental housing decreases (to QS). The difference between QD and QS represents the excess demand for rental housing at the controlled rent – there are fewer houses available than the quantity people want to rent.
It should be obvious that landlords are made worse off by rent controls - their rental housing attracts lower rents (R1 instead of R0), and they provide fewer rental housing units at that lower rent (QS instead of Q0). The producer (landlord) surplus is a measure of the landlords' economic welfare - it is the difference between the rent they receive (either the market rent or the rent controlled rent), and their costs (which are shown by the supply curve). Without the rent control, the landlord surplus is the area R0EF on the diagram, but with the rent control, the landlord surplus falls to the area R1CF. Landlords are made unambiguously worse off by the rent control.
What about tenants? The consumer (tenant) surplus is the difference between the amount that tenants are willing to pay (shown by the demand curve), and the amount they actually pay (the market rent, or the rent controlled rent). Without the rent control, the tenant surplus is the area AER0, but with the rent control this increases to the area ABCR1 [*]. Tenants as a group are made better off. However, this ignores that many prospective tenants are missing out on rental accommodation (remember the amount of rental housing has fallen from Q0 to QS). Because of the shortage of rental housing, landlords have choice over who they rent their housing units to. Given the choice between, say, a couple who both have professional jobs earning relatively high income, and a low income family who have precarious jobs, the landlord is likely to choose the former, who they probably assume to be lower risk. So, it is likely that the tenants that the rent control policy is designed to help most are likely to be one of the groups that loses from it - an unintended consequence of the policy.
Finally, the sum of the tenant and landlord surpluses in this market is what we term total welfare - a measure of the benefit to society arising from this market. Without the rent control, total welfare is the area AEF, but with the rent control this falls to the area ABCF. There is a loss of total welfare of the area BEC - the deadweight loss of the rent control. This deadweight loss arises because the gains to tenants (in terms of more tenant surplus) from the rent control are much less than the losses to landlords (in terms of lost landlord surplus).
Moreover, there are other negative effects such as landlords' incentives for keeping their housing units in good repair are reduced (because there is a shortage of housing - tenants can't afford to be choosy about the unit they rent), and the incentives to build new developments are reduced (because they will be much less profitable than if the market rent prevailed). The effects are also worse in the long run (including a larger deadweight loss). So, as Caleb Malik concludes:
...rent control is often marketed by politicians as a way to help the lower class. They paint the landlord as an old miser who is simply out to get the poor, and rent control as a way to stop this antagonist. The public often idealizes such a scenario. When not critically analyzed, rent control sounds utopian. One of our largest expenses has now been limited? Who wouldn’t sign up for that? Yet, the effects are rather tragic, as both liberal and conservative economists agree.
On the one hand, tenants find extremely limited housing, and the housing they do find is run down or comes with a price tag that is nowhere near affordable. On the other hand, landlords can no longer profit from their investments. This results in both a loss of new developments and a reduction in the quality of current housing.
[*] However, it is by no means certain that tenant surplus would increase. If the controlled rent is too far below the market rent, then tenant surplus starts to decrease. To see why, consider the extreme example where the controlled rent is zero - at that rent no landlords would offer their properties for rent, and the tenant surplus falls to zero!