It sounds like you want to measure foot-candles, which are units of light intensity received by a passive surface. This is an appropriate measure if the surface reflects light from another source. It's not the right measure for the output of a computer screen.
Also, you seem to want to measure at specific places on the screen. Here are some foot candle meters that have a detachable sensor so you can measure without throwing a shadow: http://www.grainger.com/category/ecatalog/N-1z0dwax. These are lower price models: http://www.dasdistribution.com/products/lm-economical_models.htm.
The foot candle is not an SI unit. It equals about 1 lumen per square foot. It's been replaced by the candela, which is 1 lumen per square meter. But as the candela measures light filling a 3-dimensional angular span, I think you would want to use the Lux, which measures light received by a surface.
A lumen is a measure of luminous flux, which includes only light that is visible to the human eye. If you want to measure the radiant power of a light source, you would measure radiant flux, which includes all wavelengths whether visible to the human eye or not.
Here is a better explanation and a reference for various units in which visible light can be measured, and the appropriate situations for each type of measurement: http://www.compuphase.com/electronics/candela_lumen.htm.