Hi Kaylene,
That's a good question! The amino acids are termed "acidic" or "basic" due to their interactions with water after the fact. So, when you see them as positive or negatively charged, they have already been protonated or deprotonated.
Let's now look at lysine: at a pH of 7 which is lower than the pKa of the amine group of lysine (about 8-9), the amine group of the side-chain will act as a base, donating its lone pair on the nitrogen to a proton in the solution, resulting in an overall positive charge.
I hope that helps! Another way to think of basic vs. acidic amino acids is to think of the pKas that their side-chains would generate. If it is a -COOH group, its generally going to be about 4-5, while -NH2 groups are generally going to be 8-9. As we learn from ChemOChem, the lower the pKa, the more acidic the material.
Cheers, and happy studying!
Molly