I did some research about 6 months ago about this. Since it was 6 months ago, it could be out of date now, but here goes.
The only inkjet printer on the market that printed archival quality (the ink met archival requirements - lightfast for 200 years or something like that) was an Epson and cost $800 - $900. And the replacement ink YOW!!! The other ones make no claims on archival quality ink - whether they are photo safe or not. I have an HP I use. Laser printers, since it uses a different process, are typically archival quality if I understand correctly. So for black printing that is a good thing if you have access. For color it would be great, but color lasers are VERY expensive, and not many people have access to them. If you do I am very envious!!!
There are a LOT of papers that are photo papers, but not all are archival quality, even though they may be acid-free. HP papers for example say they are acid free, but are not necessarily archival since they do not necessarily have ph less than 7.0. I know that Creative Memories has specific Inkjet and laser papers that meet the requirements.
However, I have also used some of their regular papers (not the specific ones for printers) and cardstocks to print onto. I have also printed on other papers. In those cases I mount what I have printed onto onto either other paper that is safe, or ensure that the questionable paper is not touching any photos.
Since I have an HP inkjet and am not about to shell out that much money for an archival quality one I make sure I keep the photos on a CD just in case. We have several important pictures (duaghters first days in hospital) that are only digital.
