From the outset, it is important to note that we are talking about modern photo paper here, used in modern inkjet photprinters or thermal printers, not the traditional wet darkroom photographic paper, which we offer in a different category for Darkroom enthusiasts.
The paper you choose can make a massive difference to the print quality and the purpose for which you intend to use the print.......
Very professional results cannot be achieved on cheap low quality, think paper, so you do need to consider the paper and type of finish you want to achieve. Major maunfacturers like Canon and Epson and Fuji make excellent quality papers and the premium brands sold by various supermarkets are also good value. Try to get a heavier weight of paper - anything between 100 and 200+ g, as below that, the paper weight is just the same as normal writing paper and the ink will not absorb correctly into the thin surface.
Inkjet printers can be very expensive to run and keep in ink and paper supplies - make every print count!!!
Choice of paper is also important for the purpose it is intended for - do you want to make best quality prints on archival quality paper? Or do you want cheap and cheerful glossy prints of a festive occasion to send to friends? Paper varies enormously in price and type - major supermarkets keep passable own-name brands of photo paper in matte and gloss varieties which may be good enough for your purpose, or at least for initial or draft prints.
Manufacturers of printers have their own brands of paper and your printer is likely to dowell with its own name brand as the manufacturer will have tested their printers with their papers. There are also major third party paper manufacturers of specialist phot papers , like Hannemuhle. Papers come in different textures and weights, and Hannemuhle papers for instance grace art and photo galleries and are of excellent quality for this purpose. Prints made on fine photo papers are likely to maintain their quality for many years to come. Manufacturers like Fuji also guarantee some of their papers for image quality for fifty years.
Selection of photo printing paper is a very individual thing. It is also expensive to print your photos - paper and ink for inkjet printers at best quality is not cheap. So it is always best to spend wisely and select carefully as that print will be a visual memory of your special moment for many years to come.