Is a "character song" just like a theme song for a character?
A "character song" is emphatically not the same thing as a "theme song". While there is probably no clear definition of what constitutes a "theme song", the term "character song" has a straightforward definition. That is: a "character song" is a song whose vocals are credited to the character or characters singing the song.
Typically, this means that the song is sung "in-character" - by reading the lyrics of the song, we ought to be able to glean something about the character.
Can you give me some examples of character songs?
Sure thing, person asking this question!
"Inside Identity", the ending for the first season of Chu2Koi, is a character song. Why? Because it is credited to "Black Raison d'Etre", which is a fictional group consisting of four characters from the show (Rikka, Nibutani, Dekomori, and Kumin).
But not all character songs are used as openings/endings. Some are used in-show, as insert songs. For example, "Fuwa Fuwa Time", performed by the characters in K-ON!, is a character song that is used as an insert song in some of the episodes.
Still others will never appear in their corresponding anime at all, instead being released as bonus goodies for sale. See, for instance, "you", credited to Kasumigaoka Utaha from Saekano, which never appeared in the anime.
Character songs need not be original to the anime they are used in. For example, "Tsubasa wo Kudasai" is a character song in K-ON!, since it is credited to the characters. Yet, "Tsubasa wo Kudasai" is not original to K-ON! - rather, it is a popular folk song in Japan. Nonetheless, since it is credited to the characters in K-ON!, it is a character song in that context.
Okay, so what isn't a character song?
I'm glad you asked!
Consider "Synchrogazer", which is the opening for the first season of Symphogear. It is sung by Mizuki Nana, who voices the character Kazanari Tsubasa in the show. But it is not a character song, since the song is credited to Mizuki Nana (the real human), rather than to Kazanari Tsubasa (the fictional character).
Another common misconception is that vocal theme songs may, in general, be called character songs. For example, a common belief is that "Before my body is dry" (a.k.a. "DON'T LOSE YOUR WAY") from Kill La Kill is a character song for Matoi Ryuuko. Certainly, it plays whenever Ryuuko does something awesome - but is it really a character song? No! The vocals are credited not to Ryuuko, but rather to Kobayashi Mika (who doesn't voice Ryuuko, anyway).
Well, gosh, that's a lot of information. Is there anything else I should know?
Why, yes, there is.
One potential point of confusion is the notion of an "image song". In some cases, "image song" and "character song" are used interchangeably, as in this K-ON!! album.
However, in other cases, "image song" has a distinct meaning. Like a character song, an "image song" (in this sense) is a song that is meant to evoke a "sense" of a particular character. Unlike a character song, however, an image song is not sung by the character, and hence is not "in-character". Instead, the image song is merely evocative of the character. Image songs in this sense may either be sung by a different vocalist (as with this Steins;Gate album, on which "Yakusoku no Paradigm" is an image song for Makise Kurisu), or be altogether non-vocal (as with this Accel World album, on which "re-Incarnate" is an image song for Arita Haruyuki).
The Tokyo Ravens album "Reincarnation" is another example that's full of image songs in the latter sense, with vocals credited to Kurosaki Maon (who does not play any of the characters in the show).
I find that image songs (in the latter sense) are relatively rare as tie-ins to anime. They do appear rather frequently as tie-ins to visual novels, though. I'm not sure why.