As Salaamu Alaykum,
more detailed answer
Whilst praying Salaah it is recommended to cover one's body properly and as much as possible. However, there is a minimum requirement as apposed to the recommended amount of the body we must cover.
Covering the elbows is not necessary but it is surely recommended that one covers their elbows whilst praying Salaah.
difference in a T-Shirt and Shirt...
When one is wearing a T-Shirt it is taken that he cannot cover his elbows (not that he doesn't have the means generally, but at that time he cannot cover them), therefore, Salaah will be done in the T-Shirt. However, when one has a shirt on then he has the means to cover his elbows, therefore, if he does not do that, and leaves them rolled up, it is disliked because despite having the means to cover one's elbows he did not (and Salaah will still be valid).