If you design a logo in Photoshop and send it to a client as a JPEG, you are doing them a massive disservice. Here is why you must understand the difference between Vector and Raster graphics.

Raster (Photoshop)

Raster images are made of pixels. Photographs are rasters. If you take a 100x100 pixel image and scale it up to a billboard, it will look like a blurry mosaic of squares. This is terrible for logos.

Vector (Illustrator)

Vectors are based on mathematical equations. A circle in Illustrator is drawn using math, not pixels. This means you can scale a vector file up to the size of the moon, and it will remain infinitely sharp and crispy.

Conclusion

Always design logos, icons, and typography in vector formats (.AI, .SVG, .EPS). Use raster for photos and digital paintings.