In the Tone Curve panel in the Develop module, click on the point curve icon, circled below:
This gives you a version of the curve with actual points on it. You can click and drag on the existing points, add new points by clicking on the curve, and remove points by clicking and dragging them out of the curve graph completely.
If you find a look that you really like, save the tone curve settings as a preset. In the Presets panel (left in Develop), click on the + next to the panel name, give the preset a name, put a check mark next to Tone Curve, uncheck everything else, and click create. Select another image you want to apply the preset to, click on the preset name in the Presets panel, and voila!
Of course the point curve can be used for “serious” photographic work as well — it allows you more precise control over the tones in your image than the default parametric curve does. See my Introduction to Curves post for an explanation of how to read and use the curve.