Why is the Female Titan after Eren?

Question

As shown in episode 17, the Female Titan is after Eren, looking for him in the formation outside the walls. Why is the Female Titan after Eren?

Answer 1

This answer contains major unmarked spoilers if you've only seen the anime.

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As you've seen in the series, certain titans have abilities, such as Annie's ability to harden certain points during combat, or the Armored Titan's ability to harden his entire body to smash through solid structures.

Eren too has an ability like this, and given the nature of the world, it's very valuable to control. It's called the 'Coordinate Ability', and has three separate sub-abilities currently revealed - Eren, however, has currently only demonstrated the first:

First and most noticeable of all, the Coordinate power grants the user the ability to control the Titans at will and make them follow virtually any order. ^{[Ch. 50 (p. 35-44)]}

Secondly, the Coordinate allows the user to erase or modify the memory of a single person or even those of entire countries if they are skilled enough. ^{[Ch. 63 (p. 8), Ch. 64 (p. 37-38)]}

Finally, the power allows the user to inherit the lost memories of mankind and those of the previous users, giving them omniscient-like knowledge about the world, such as the reason the Titans exist and how the Walls were created. ^{[Ch. 64 (p. 39-42)]}

_{Coordinate on the SnK Wiki}

Furthermore, there is an answer to another question here which gives some more information on the ability itself.

The Titan Shifters that infiltrated the walls (Annie, Reiner, Bertolt) have the goal of finding and acquiring the coordinate ability, and this is the reason why Eren is kidnapped by the latter two around chapter 45.

Assuming a second kidnap ends successfully, it is unsure what the other shifters would do with Eren. Due to the fact they haven't attempted to gain his ability through eating him already, instead but opted for kidnap, it's likely that they'd prefer to keep Eren alive and attempt to either persuade or coerce him into cooperation with their cause. What plans they have for the coordinate ability after this point is also unrevealed at this time.