300 dpi is usually a good start for sending it to the printer. But I've heard different printers say different things about the format.
Some want PDF, others like TIFF, some are OK with JPEG. I would save it as a regular PSD and then ask the printer what he needs to get things going. You can always export the original PSD to whatever, It's a lot more difficult, to get things back to square one.
As for Vectors. Vector Graphics are based on lines, points, curves and such that relate to each other to make up a graphic. Vector can be scaled up and down without degrading the image. This is opposed to raster images like bitmaps, gifs, jpegs and the like which are a grid of colored pixels. the more you scale them the more corruption in the image.
Adobe Illustrator, Flash, Freehand, CorelDraw are Vector programs. Photoshop, Painter, MSPaint are for Raster images.