I like to think about the tone curve in terms of a function:

y = t(x)

The input value x to the function is a pixel value from the source image, which is actually the brightness of that pixel. Let's say the values of x can range from 0.0 (black) to 1.0 (white), with all the gray tones represented as real numbers between 0 and 1 (to simplify things a bit let's assume the picture is black & white so there is only one value per pixel, I will later extend the concept to color pictures).

The output value y returned by the function is the transformed pixel value, also in the range 0 to 1.

Thinking in these terms, a tone curve transformation is done simply by calling the t() function above for every pixel in the source image and replacing that pixel with the transformed pixel returned by the function.

If you have a color image, then there are a couple of ways you can go about it. You could apply the function to the "L" component of every pixel using the HLS color model. Another option would be to apply different tone curve functions to R, G and B, then assemble a transformed RGB image with all those results. In Lightroom you only have the first kind, or in other words, a single curve for the whole image. In Photoshop or GIMP, the Curves adjustment can also be applied independently to the R, G and B channels.

A natural way to represent this t() function is with the use of a curve. Just represent the x values along the X axis and the y values along the Y axis. Then for each possible value of x you plot a point at coordinates (x, t(x)).

The simplest possible curve would be a diagonal line, going from (0,0) to (1,1). This would be when you are not applying any transformation at all, or in other words, t(x) = x.

A curve that is a horizontal line at the Y=1 height would mean that for any input the result is 1, or white. This would transform any image into full white. Conversely, a horizontal line at Y=0 would make any image fully black.

If you wanted to increase the tonal range available for the midtones in your picture, then you can do so by compressing the darks and the highlights. The shape of the curve in this case will resemble a slanted letter "S".

As a general rule, if a segment of the curve is more horizontal than vertical it means that you are compressing the range of tones covered by that segment. If the segment is closer to a vertical then the range of tones is being expanded. A segment that is a perfect diagonal means that there is no expansion or compression, just a linear mapping from one set of values to another.

One additional comment. While you can edit the Lightroom Tone Curve directly by dragging the curve, the tool has these four sliders below the curve that I think are pretty cool. Instead of messing with the curve directly, the sliders are pre-programmed to affect four different sections of the curve independently. By default each slider affects a 25% of the tonal range space. The little buttons over the X axis of the curve can be dragged left and right to change how these four regions are defined. I find I get better results when I modify the curve via the sliders, for me it is easier to think about the adjustment I want to make in terms of just four areas.

Edit: Regarding compression and expansion, consider that in a picture there are a finite number of brightness values that can be represented, so you have to distribute these tones in the way that benefits your picture the most. For example, if your picture came up too dark, that means that a lot of the pixel values are going to be on the low end of the brightness scale, and the brighter part of the scale will be a lot less used or not used at all. With the Tone Curve you can expand those dark tones to cover a larger part of the brightness scale, and that can make stuff that was barely visible in the original much more clear to see. Since the brightness scale is finite, if you expand in one part you must compress in another, the curve just allows you to do localized adjustments in different parts of the scale, so you can sacrifice (compress) in one part so that you can provide more tonal detail in another.