I'll start with some basics the way I understand them. Please feel free to correct me where wrong and add as you see fit-

Di-chlor. Granular chlorine with stabilizer added. Pros- Dissolves quickly, low impact on PH. Cons- Repeated use causes stabilizer build up resulting in more chlorine needed to do the same job or a partial draining and refilling to lower the stabilizer total.

Tri-chlor. Can be granular or pressed and formed (sticks and tabs) chlorine with stabilizer added. Pros- Fast dissolving in granular form, slow in pressed and formed which can be added to skimmer baskets or chlorinators to slowly dissolve and add chlorine to the pool. Cons- very acidic, can effect PH levels, same stabilizer build up issues as dichlor.

Calcium Hypochlorite- Usually granular chlorine with added calcium. Pros- Doesn't contain stabilizer so no build up issues. Added calcium can protect plaster etc. pool walls and floor. Low impact on PH. Cons- Very slow dissolving, can damage vinyl liners. Calcium content can add to TDS and cause hazy water until filtered out. Filtering out can cause a hard build up on DE filters that makes a breakdown and manual cleaning necessary, causing pool downtime (personal experience).

Lithium Hypochlorite- Granular chlorine. Pros- Very fast dissolving. Low impact on PH. No added stabilizer. Cons- Most expensive source of chlorine available.

Sodium Hypochlorite, aka Bleach- Liquid Chlorine. Pros- In liquid form so it's pre-dissolved. Low impact on PH. No added stabilizer. Comes in varying percentages such as 3%, 5.25%, 10% and 12.5%. Least expensive form of chlorine available. Readily available, can be purchased at grocery and department stores (yes the same "laundry bleach" is the same as the pool store "liquid chlorine", sometimes the percentages vary is all. Check the label to adjust quantity necessary to do the job). Cons- Disposing of plastic jugs. Pool store owners frown at you.

Something like that.