Third-party ownership ("TPO") is a mechanism through which an investor, whether an individual or a corporate entity, can purchase the economic rights of football players, thereby making a profit each time "their" players are sold.
Although at the time the practice was fairly widespread in some European and South American countries, TPO came to most people's attention in England in 2007 when West Ham purchased Carlos Tevez and Javier Mascherano through a network of investment funds at a considerable undervalue. The following season, these two players effectively combined to save the Hammers from relegation to the Championship, a disaster that would have cost the club heavily in TV rights, sponsorship deals and prize money. The teams that did get relegated that season cried foul.
The next season, the English Football Association outlawed TPO and, in 2015, FIFA followed suit rendering TPO unlawful all over the footballing world. New FIFA President Gianni Infantino spoke out on the issue, calling TPO "akind of modern slavery, where you see players belonging to investment funds, or other, generally unidentified, corporate entities".
He's not wrong - the third party owns the player's economic rights instead of the Club for which he plays owning his registration. That third party needs to monetize its asset and the concern over TPO has always been that the interests of the player and the club are lost when this happens. The human elements of the beautiful game; the player's affection for the fans, the style of football the club plays, the big wins over the arch rivals - these are all lost on the third-party owner whose interest, it is said, is driven by the player's retail value, not their ability.
So the player, be they an international superstar or a Peruvian unknown, end up being traded like cattle and neither they, nor the buying or selling club, have any meaningful say in where they go, when or for how much.
Does this create an unfair environment where clubs can gain a competitive advantage against others by purchasing players at an undervalue? Many would say yes. Does this mean that a player may end up being sold to a club on a different continent against his wishes? Clearly that is a distinct possibility.
On the flip side, there are many clubs who argued against the ban on TPO on the basis that they maintained that it was the only realistic way in which they could compete in a sport where transfer fees and wages are escalating ever further.