For people who know the terminology, Andrew and Jeff explain this well, but it's a difficult thing to explain to people new to soft-proofing.
Consider this version: your monitor, even if calibrated and profiled, will not ever display color exactly the same way that the print will, because of the inherent differences in transmissive light (monitor) vs. reflective light (ink on paper).
The range of colors a device can reproduce is its gamut, and the image file itself has its own gamut: the color values it contains. Some of these colors can be accurately printed, others accurately displayed on the monitor, but not necessarily both.
Soft-proofing in Photoshop uses your monitor to simulate a visual interpretation of what the image will look like when printed on a specific printer and paper. So you're simulating the image gamut, mapped to the printer gamut, mapped to the monitor gamut.
When soft-proofing, the black ink and paper white simulations will usually look awful on the display. But this is reality, because the color management system is translating the color values of the file through the color gamuts of the device profiles. Blacks on a monitor will appear much darker than a print because on a monitor black is the absence of light. On a print, even black ink is reflecting some light.
You have to train your eye for Soft-Proofing; especially for black ink and paper white.
And you have to use the Simulate Paper White and Black Ink options to get it right.
Put two copies of the same image side by side (use the Image > Duplicate command) - one is soft-proofed, the other is not - the goal is to visually match the Soft-Proofed version as closely as possible to the Reference (not soft-proofed).
Use adjustment layers to make them match, usually a boost in saturation (maybe around +10 in Hue & Saturation?) and Curves (usually lightening the shadows to open them up). With some papers (eg. Silver Rag, Premium Luster) you won't have to make much of an adjustment. With others (eg. Somerset Velvet, PhotoRag or any kind of canvas) the adjustments will be more extreme because of the limited color gamut of the substrate.
In my classes I've been asked the question "but since we're using a custom profile for the printer and paper, doesn't that take care of the color matching?" No, because of different color gamuts. Remember that the photo has its own color gamut (depending on the working color space) and some of these color values can't be reproduced exactly with the chosen printer/paper combination. The numeric color values are translated through the color management system and always have to be adjusted for the gamut of a specific device.
Within Photoshop, files with embedded or assigned working profiles (Adobe RGB, ProPhoto, sRGB etc.) are always displayed color-managed. If you use a good monitor calibration system (EyeOne, not Spyder), you can trust the colors in the Master file. But you still have to adjust for the output.
So when soft-proofing, you will almost always need to make adjustments to your Master file to get the print to match what you see on the screen. I have Master files with Adjustment Layers made for Museo Silver Rag, Epson Premium Luster, Hahnemuhle Photo Rag, Breathing Color Canvas... the list goes on and on. Each printer and paper combination has been soft-proofed and adjusted before printing.
And in the end, sometimes there is no substitute for a hard proof. Be willing to make test prints and make adjustments accordingly.