Please explain this economic term in plain, simple and understandable words, so even a wretched fool like me can get what is meant by EPS.
How do EPS help one pick up a "better" stock? What do they indicate? Is the number shown together with any stock chart by itself helpful or does one have to put it into perspective?

To be honest, I don't think I understand the definitions of EPS I just read on the web. They do not help me picture the concept of EPS and, what's more, they do not even explain why EPS are sometimes negative.

The net income of a company is the profit a company makes after paying all expenses and taxes.

Shares outstanding is the total number of shares that a company has.

If you were to take the net income over the last 12 months and divide it by the number of shares outstanding, you get the EPS number.

So if the Net income was divided equally among all the shareholders, the number you would get is the EPS. Sometimes a company's net income is negative, This is why EPS is negative. That means that expenses were more than the revenue the company brought in.

The EPS does not help you pick a "better" stock, it's just one of many measures of profit.