In a “condo” arrangement, you legally own a particular unit in a multiple unit structure of the building.  Under a typical arrangement, you have a share and a right to use common areas such as hallways, elevators, gardens, swimming pools, and club house within that structure.  You pay monthly payment to an “association” for maintenance expenses the common areas. The association is typically run like a corporation with complaint and appeal processes to protect individual rights of owners and to provide a mechanism for resolving disputes within the community.
In a “co-op”, the ownership structure is quite different: you do not own your own specific unit in the building but own stock in the corporation that actually owns the building and all the apartments. You lease your apartment from the corporation according to a formula based on the unit’s size.  As a shareholder, you have a say in electing the Board of Directors who manage the cooperative.