I'm reading the news about a company and they state that: "net profits attributable to the shareholders jumped to US $25.6 million, or 15 cents a share". What does it mean in terms of share price? Should the share price increase by 15 cents?
It means that the company earned 15 cents per share in the most recently reported quarter.
Share price may or not be affected, depending on how buyers and sellers value the company. Just because profits "jumped," does not mean the shares will follow suit. An increase in profits may have already been priced into the stock, or the market expected the increase in profit to be even higher.
As the shareholder, you don't actually get any of these profits into your hands, unless the company pays out a portion of these profits as a dividend.
It's a way to help normalize the meaning of the earnings report.
Some companies like Google have a small number of publicly traded shares (322 Million). Others like Microsoft have much larger numbers of shares (8.3 Billion).
The meaning depends on the stock. If it's a utility company that doesn't really grow, you don't want to see lots of changes -- the earnings per share should be stable. If it's a growth company, earnings should be growing quickly, and flat growth means that the stock is probably going down, especially if slow growth wasn't expected.
What does it mean in terms of share price? Should the share price increase by 15 cents?
No, but you're on the right track. In theory, the price of a share reflects it's "share" of time discounted future earnings. To put it concretely, imagine a company consistently earning 15 cents a share every year and paying it all out as dividends. If you only paid 25 cents for it, you could earn five cents a share by just holding it for two years. If you imagine that stocks are priced assuming a holding period of 20 years or so, so we'd expect the stock to cost less than 3 dollars.
More accurately, the share price reflects expected future earnings. If everyone is assuming this company is growing earnings every quarter, an announcement will only confirm information people have already been trading based on. So if this 15 cents announcement is a surprise, then we'd expect the stock price to rise as a function of both the "surprise" in earnings, and how long we expect them to stay at this new profitability level before competition claws their earnings away. Concretely, if 5 cents a share of that announcement were "earnings surprise," you'd expect it to rise somewhere around a dollar.