In basic terms, what Barmaid said refers to certain rights that people can accrue even where there is no legal ownership "on papers", a good example is when a couple split who have been married for years, the husband may have been the breadwinner, the house was in his name only etc... but the wife contributed to the household by keeping the house, raising children etc... so the wife may have some claim in equity over the marital home as they have both contributed to the household.
It is the difference between legal ownership (eg on paper such as having the title) and beneficial ownership, the latter is based in what is called equity which is, essentially, rooted in fairness but there are rules governing this.
Fairness applies to the end result too so there is no unfair gain, eg getting significantly more than it is ruled that you contributed.
When there is a beneficial interest, the legal owners are said to hold the beneficial interest in trust for those with the beneficial interest.
I hope this is right and helps, long time since I studied this! Hopefully Barmaid will appear to explain better.