"Tier" would be an animal of sorts. "chen" is a common suffix for cutiefying something or indicate it's small (and adorable).
"Tierchen" would be a cute animal, usually a small one too.
The suffixes "-lein" und "-erl" function similarly but have been largely replaced in common german (though in southern germany, both are alive and well)
erl ... would be more related to the actual size
Sack -> Sackerl (bag -> small bag)
Wagen -> Wagerl (cart -> small cart)
lein ... would be more related to the inner size (sorry I can't frame this into words any better) Frau -> Fräulein (Woman -> Younger/Fragile woman (unmarried))
Wagen -> Wäglein (Cart -> smaller but also lesser cart)
edit: referring to the answer of pavel_lishin. Yes "qualitative" modifier was the word I am looking for.