Summary (or, "But I didn't sign up for the organic chemistry lecture!"):
- The numbers are the amounts of major plant nutrients.
- Get a "balanced" (numbers that are close to each other) fertilizer, preferably "with micronutrients".
- Read the label.
- Mix and apply as directed.
- Liquid Growth Indoor Plant 4-12-9 Food;
- All Purpose Liquid Concentrate Plant Food 12-4-8 with micronutrients;
- All Purpose Liquid Houseplant Food 8-7-6.
Do the numbers have any significance? If so, what do they mean?
The numbers represent the percentages by weight in the fertilizer of the major plant macronutrients. In short form we call them "N-P-K" which are the chemical symbols for Nitrogen, Phosphorus, and Potassium, respectively.
The amounts of the latter two don't represent the pure element, but instead P2O5 and K2O (or "the amount of oxide in the form of P2O5 and K2O that would be present in the fertilizer if all the elemental phosphorus and potassium were oxidized into these forms" (source)). It's confusing, but this doesn't mean that the P and K present in the fertilizer is actually in the form of P2O5 and K2O!
So the first fertilizer (4-12-9) is 4% Nitrogen by weight, 5% Phosphorus by weight (0.436 * 12%), and 7% Potassium by weight (.83 * 9%). For purposes of feeding a houseplant, you don't really care about this. But if you're gardening outside and get a soil test back from the lab, this is a similar formula to how they calculate the recommended fertilizer analysis for your needs.
There are a few critical things to be aware of when selecting a fertilizer:
- Not all of the N-P-K that is present based on those numbers is "available" to the plant. There is usually a note on the back of the package that tells you what percentage is in a form that the plant can use. (If you have high analysis fertilizer, it is often beneficial to have a smaller portion that is available to the plant immediately -- the effect is "slow release" so that more of the fertilizer becomes available to the plant over time.)
- These three nutrients aren't the whole story when it comes to plant nutrition. The fertilizer that says "with micronutrients" will contain several other elements that are required in smaller amounts but are still essential to plant health. (E.g. calcium, magnesium, sulfur, etc.)
- Larger numbers (high analysis fertilizer) aren't necessarily better. Too much nitrogen can "burn" plants -- possibly killing them. Too much of other nutrients can cause imbalances. For example, too much potassium can make it so that the plant can't take up nitrogen, magnesium, and/or manganese.
Without knowing exactly what it is you're feeding, my generic advice for a generic office plant would be to get the "with micronutrients" fertilizer and apply according to the directions on the label.