Vector-based files will allow you to scale your logo without losing quality and getting pixelated, making these files best suited for print design. EPS is the exchange format between Illustrator and CorelDraw or older Illustator versions.
↓ Illustrator AI
↓ Adobe PDF
↓ Postcript EPS
↓ Full Suite
High res images are used for printing purposes that don't need scaling and high res video. Low res images are used for web applications or low res video. PNG and TIFF formats have transparent backgrounds, which means they won't have a white rectangle behind.