There are three categories for donation: organs, tissues and eyes.  Each has its own set of requirements and "rule-outs" (situations and conditions that make donation not possible).

Organs: Heart, lung, liver, kidney, pancreas, intestine
These organs can only be procured (surgically removed) from what is commonly known as "brain dead" donors; those donors whose heart is still beating but yet they have no brain activity.  A common cause of 'brain death' is a head injury due to a vehicle accident.

After the donor is taken to surgery, anesthetics are given and life support is withdrawn allowing the donor to pass away.  The procurement surgery is performed seconds later.

The procured organs are then immediately transplanted into the recipient

Tissues: Bone, skin, tendon, fascia, vein, heart valve
Unlike organ donors, tissue donors do not need to have a beating heart.  In fact, tissues may be procured up to 24 hours after the donor passes away. 

Also unlike organ donors there is no need for the donor and recipient to be matched based on age, size, blood type and so on.  It is for this reason that people from only weeks old to even a century old can donate tissues.

After procurement tissues are treated with various antibiotic processes and shaped and molded into the final transplantable tissues.

Eyes: Whole eyes and corneas
Eyes and corneas are similar to tissues in that they can be procured from non-heart beating donors but also similar to organs in that age, size and medical/physical conditions do matter.