TDS used in this context means "Total Dissolved Solids" and is a measure of the amount of nutrient salts that are present in your nutrient solution. The amount of total dissolved solids gives an indication of water purity as well. Distilled water should have a very, very low TDS (0 to 10 ppm). Reverse Osmosis water will be slightly higher in TDS (removes 99% of impurities). Tap water will be quite a bit higher in TDS (typically 250 to 700).
Total dissolved solids are typically measured in Parts Per Million (PPM) - but can also be measured using the electrical conductivity (EC) of the solution. Most TDS meters actually do measure the EC and then convert this reading into an approximation on the PPMs.
So what are you measuring? Nutrients Salts. Or better yet, ions with a positive charge. The more ions with a positive charge, the more nutrients are in the mix. Note that a TDS meter does not tell you the composition of the salts (how much of N, P, and K are present, for example) - only the total number.
Because TDS is a measure of a solute dissolved in a solvent - it follows the rules for dilution. So if you mix your nutrient solution a little bit strong - you can add water to get the correct strength (dilution). If you mix it a little bit weak, you can add additional nutrients. In this way, you can get whatever nutrient strength (as measured by PPMs) that you desire.
The correct PPMs to use at every stage of growth is hard to specify. It depends on many, many different factors. But in general you can start a seedling out at about 250PPM and if you are running CO2 in a flowering environment the plants can handle as many as 1500PPMs. I have personally run flowering plants of certain varieties, under CO2, as high as 1750 with no damage.
If the nutrient solution is to hot (the PPMs are too high for the stage of growth) - then you will "burn" the plant. This is where the leaves start to curl down and die at the margins. When you see this, you should immediately dilute your nutrient solution (though this is the same symptom for being too close to the light - so check both). If the nutrient solution is too weak, then you will get stunted growth - which is hard to detect without experience. If you have an incorrect ratio of nutrients (too much N and not enough K) - then you will get a single element deficiency.
Hope this helps and