Remember the italics! "For him?" Meaning, "Not him, Lily." Also, the next lines are Dumbledore: "After all this time?" meaning the fact that it had been 15/16 years since she'd died, did he still love her? And Snape's reply: "Always" Basically because he is lily's son.
You are missing easily one of the greatest literary love stories if you don't understand this paragraph in the book. Snape's Patronus was a doe, which was also Lily's Patronus. Snape's doe even had Lily's eyes. Snape was also the only Death Eater capable of producing a Patronus because of his love for Lily. The person above is correct in stating that Snape was stating that it is not Harry he cares for, but that he has always been in love with Lily. If you'll remember, after Lily died and Dumbledore confronted Snape, Dumbledore made Snape promise to protect Harry. His love for Lily bound him to this promise, and his Patronus' likeness to Lily's was what made Harry feel like it was familiar and led him to the Sword of Griffyndor.