The main strength to Eidolons is raising the chain gauge while being able to
avoid damage and
ressurect your party (When Eidolons leave you are left with full health even on previously KO'd members).
As identitycrisisuk said,
If you summon an Eidolon during a fight with an Adamantoise, It will cause the legs to die, which makes it fall on its face. During this phase it cannot attack, due to the difficulty of these monsters as well as the great CP they give (40K) and chance to drop Platinum Ingots (Sells for a ton) and Trapzohedron (Used for Crafting). It has become a very popular practice to start the battle off with a summon to make it much easier (Certain strategies are used to kill it before it gets back up). So this is one such enemy that answers your question.
I had trouble with a few random encounters through out the game and found that waiting until i was about to die, then summoning an eidolon was just the right move that needed to be done to stagger the tough enemies and eventually lead me to victory. TP is very easy to get in this game so i don't really consider this a waste.
Bahamut is most likely the first way you will be breaking damage limit early in the game, and
Hecatoncheir (Vanille's Summon /spelling) is good for inflicting status effects while you can buff yourself or spam death.
It should also be noted that everytime you get a Crystarium Upgrade, Your summons get stronger (Maxing out at the last Crystarium Upgrade, after the final boss).