Your monitor produces a picture by transmitted light using glowing phosphors and the red/blue/green colour system. A paper print uses reflected light from a picture printed using dyes or pigments and the cyan/magenta/yellow system. It is not surprising that the results look different. However, software does attempt to make the conversion from one to the other and can get close given ideal conditions.
Photoshop can make this conversion and so can your printer software. The most important thing is to ensure that only one of these is doing it. A double correction is just as bad as none at all.
The whole business of colour management is a real can of worms. However, if you use the settings shown below you should get good results.
With other software try to switch colour management (correction, matching or whatever it is called) OFF in your imaging software. Switch it ON in your printer software (see 5 below).
If all else fails, you will have to do what I do, which is trial and error - see method 2 below.
1 Make sure that you view your screen in subdued light. View your print in daylight.
2 If you have a CRT screen, set it up using the Adobe gamma utility. This comes free with Photoshop and can be found in the Windows control panel. Adobe gamma is NOT suitable for LCD screens, which should be used in their 'Standard' or 'sRGB' settings.
3 In Photoshop Elements go to Edit > Color Settings, and select 'Always Optimise Colors for Computer Screens'.
In full Photoshop (and possibly other software) the settings are more complicated, see below:
4 Ensure that Photoshop does NOT perform colour management. Select File > Print. On this screen select 'show more options' and ensure that the Printer Profile is set to 'Printer Color Management'.
5 Click 'Print' and then 'Properties'. The next screen will depend on your particular printer but the usual thing to do is to select 'automatic' and the correct grade of paper. On my printer the screen appears as below:
6 All these settings should be saved and will automatically be selected whenever you come to print an image.
Also note:
7 When saving an image, use 'Save as' and ensure that the 'ICC profile' box is checked. This ensures that the correct settings are saved with your image.
8 When loading a new image you may get an 'Embedded profile mismatch' box. Select the option to 'Convert document's colors to the working space'. This ensures that the correct settings are applied to any new images you load.
Crop a small section of your image. I find a vertical or horizontal slice works best. Try to ensure that it contains an example of all the important colours in your picture (e.g. skin tones).
Print that and examine it. If it is OK, then go ahead and print the whole picture.
If it is e.g. too light, then darken the whole image using Image > Adjustments > Curves, and try again.
If it is e.g. too green, then reduce the green by using Image > Adjustments > Variations.
You may need to do several experiments before you get it right but it will get quicker with experience.