When you are raising a tama the attention icon comes on when one of the basic needs (feeding / discipline / bonding, etc.) either becomes empty or needs boosting.
(I can't be more specific because there are different care needs for different versions.)
But basically when the attention icon comes on for something, you need to check what the tama needs. If you do not care for your tama's needs within a certain amount of time - sometimes 10 mins, sometimes 15 minutes (again, depends on the different tama versions), the attention icon will switch off.
That's a "care miss" and these do usually affect the kind of character your tama will evolve into.