Yes. If you kill ships belonging to a faction then your standing with that faction will go down and your standing with the enemies of that faction will go up.
Factions include the four player factions (Minmatar, Caldari, Gallente and Amarr) as well as pirate factions such as Blood Raiders, Serpentis and Sansha plus other NPC factions like Concord, Sisters of Eve, EoM and so forth.
Shooting Caldari ships will make the Caldari state like you less but the Gallente and the Minmatar will like you more as a result.
You don't really need to worry about it unless your faction drops by a lot (to less than -2) t that point you may start getting shot at by faction navies in their space. The Diplomacy skill will help to shore up your faction standings with people who don't like you also, you can decline missions against factions you don't want to lose standings with as long asyou don't decline more than one every four hours.
