TL;DR Only you can buy houses on your turn. All players can buy houses between the turns. So between turns, there could be a house auction.
The rules about building houses seems to be the subject of much debate. Everyone I know builds their houses on their own turn. If you do that, then there will never be two people trying to claim houses at the same time. Unfortunately, the rules don't say you can only build houses on your own turn. They explicitly state:
When you own all the properties in a color-group you may buy houses from the Bank and erect them on those properties.
[...]
Following the above rules, you may buy and erect at any time as many houses as your judgement and financial standing will allow.
Nothing about your own turn there. Looks as though you can just build whenever. Here's another one that implies that you can buy houses simultaneously:
When the Bank has no houses to sell, players wishing to build must wait for some player to return or sell histher houses to the Bank before building. If there are a limited number of houses and hotels available and two or more players wish to buy more than the Bank has, the houses or hotels must be sold at auction to the highest bidder.
This is backed up by some tournament rule pages:
The buying and selling of houses does not have to occur on your turn, but rather, can be done at any time during the game.
The largest concern here is that someone can buy a house between someone rolling the dice, and landing on a property. Wikipedia says the following (even though it doesn't have a citation for it):
When a player owns all of the properties in a color group and none of them are mortgaged, they may develop them during their turn or in between other player's turns.
I tried to find another source for this and found it here, though again this is not an official source:
The way the rule is applied in tournaments is that it means that you do not have to wait for your turn to buy houses. They allow you to buy houses between player’s turns. Once a person has started their turn by rolling the dice you must wait until the end of their turn before you can build. What we typically do is as the next player is reaching for the dice we call out “Hold the dice, I want to build”.
This means that between players' turns there can be an auction between various players that want to buy the same houses.
These auctions can proceed as you see fit, but are managed by the banker. This page is rather thorough in its explanation on these auctions:
The Banker runs auctions. Players must indicate the specific property to be improved if the house/hotel is won before bidding. Auction bids start at the value of houses/hotels for the lowest-priced property where it might be placed. The highest bidder, after giving his/her money to the Banker, places the house/hotel on the identified property. If houses/hotels remain and more than one person wishes to purchase the remaining houses/hotels, then another auction is held for the right to purchase these houses/hotels, one by one.This continues until no houses/hotels remain, until no one wishes to buy, or until only one person wishes to purchase houses/hotels. If only one person wishes to purchase, he/she may do so without an auction.
