To be honest, I find the system of photometric units rather confusing. This is because there are several aspects: Spherical or plane, physiologically weighted or not, lamp or illuminated object side. You can find precise definitions of these units in Wikipedia, start with the lumen entry.
I'll try to do a very simple explanation (I know it's oversimplified! Don't complain, read the Wikipedia articles). Let's start with the lumen. Lumen measures the total amount of light, independent of distribution (focusing). If we'd measure water, it would be like the liter. Watering your lawn with 100 liter of water specifies just the amount, not more. Candela is lumen per (spherical) solid angle - you could say intensity. Going back to the lawn, it would be like l/m2. If you lawn is 100 m2, then you get 1 l/m2, if you put 100 l into a plant pot, then it may be 10000 l/m2. With optics, this would be focusing, concentrating a certain amount of light on a smaller angle. From a light source with constant light output, you get higher intensities (in candela) by focusing, reducing the light angle twice (eg. 10° to 5°) gives four times the intensity (as the area is reduced by four).
Lux is lumen per square meter - in our lawn analogy also like l/m2 (I said it's oversimplified). This measure is commonly used to qualify object side lighting, like how bright does your office desk need to be or how bright does the full moon shine? "Illuminance" measured in lux is useful for comparing lamps only if you have identical measuring conditions (distance = 1m).