Most images made up in graphic design programs or from digital cameras are made up of pixels; these are called Bitmap images – a grid of pixels or bits.
When you increase the size of a bitmap image, you cannot create more pixels; although additional pixels will be generated by the software, they will only contain approximated data. This can result in pixellation.
Although print machinery can reproduce better resolutions than 300dpi for many reasons – diminishing returns on quality, file size causing issues with data transfer or rendering speeds, its considered better practice to use vector imagery.
Vectors, as in the top example above, are infinitely scalable forms. This means they are especially useful when creating very large documents; fonts can be retained as vector images and stay perfectly sharp even on bus sides or street posters/ad boards, where the background may be less focused, its also viewed from further away so any pixellation exhibits itself as natural blur.
Some artwork may be created completely from vectors, for example Illustrator documents are usually vector artwork for the main part. In other desktop publishing programs fonts may be the only vector item. Vector versions of logos are indispensable as they remain sharp for sign writing, and allow the paths to be traced for embroidery, carpet logos or light goboes.
PDFs contain vector imagery, usually fonts but can be more. A good starting point for vector artwork is to start saving Photoshop PDFs (you can allow it to discard layers, and the font vectors and any Smart Objects used will be retained). These can often be quite bulky but if you use a program such as Dropbox or WeTransfer to send us your files, thats no problem.