I would like to print something on A1 size paper, and just opened my Photoshop to edit the picture and I am confused about somethings. I found online which size to choose, but other than that all the dpi/resolution is confusing me. I just need to print a full colored picture picture on a poster so, I'm confused if I am doing it the right way.
You are in the wrong Photoshop menu. You are in the menu to create a new blank image, but you already have an image. Open your image. Then you should be in Photoshop menu Image - Image Size. Uncheck the Resample box at bottom, you only want to scale dpi for printing. Your A1 paper is 841x595 mm, select mm and enter that (Document Size). For your given image size, whatever size it is, it will show the resulting dpi calculation (image dimension in pixels / paper size in inches) The tool will convert from mm if you select mm. A1 paper likely will not be near 300 dpi, maybe 150 dpi (no clue what size your image is). But that is how it is, that is how large your image is, and large wall posters do not need to be as much as 300 dpi.
You are probably not printing posters at home. If you are sending the image out for printing, you don't need to do this, just give them the image and tell them paper size and they will do it. Except you should verify the calculation first to ensure you have sufficient pixels to print this large. Or to do that, you may want to first tell your print service the size of your image (dimensions in pixels) and discuss printing A1 paper for their advice.
There is more to it, the image must also be cropped to be the same shape as the paper shape. The A1 paper is metric aspect 1.414:1, and the image is some different shape. See http://www.scantips.com/lights/resize.html
This is a dialogue that creates a new image and the DPI shown here is only for reference if you want to set the resolution of the document according to how large you want to print it. However, since you already have an image (that you're not creating from scratch), you can open it directly (you don't need to create a new document for that) and only be concerned about DPI when you are ready to print it.
DPI is a physical measure meaning how many pixels of the image will go into one linear inch of paper; it is a measure of density. 72 PPI is a somewhat antiquated value that many displays were assumed to have some decades ago. Usual print settings today are 150 or 300 DPI.