A bladder is a flexible sac which can expand and contract and which holds liquids or gases, especially the urinary bladder. The bladder is also the term for the inflatable bag kept inside balls used in sports, such as soccer balls and Rugby balls.
Think of a bladder as a holding sac. What is inside can be released as needed. Two examples in the human body are the gallbladder which holds bile until the body takes in fat. The bile makes large fat molecules into much tinier lipid molecules so that enzymes called lipases can break them down and the body can absorb the lipids. A person who has their gallbladder removed and eat a meal with a lot of fat in it, will find that much of the fat will pass right through them.
The other bladder you are more familiar. The Urinary bladder.
an organ that stores liquid wastes from the kidneys temporatily
A bladderful is an amount of something which can be held in a bladder.