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.
Expectations vs new information
When the market acknowledges new information, the expected impact on the share price represents the difference between that information and what "the market believed" before that, which was already reflected in the share price.
Before the profit data was released, what did people think that this company net profits would be? If they already thought that the profits would be $25m-ish or 15-ish cents a share, then these news would just confirm it and the share price should stay the same; if they for some reason thought that the profits would be larger than this, then this is bad news and the share price should fall, and if (as the "jumped to" phrasing implies) they previously thought that the profits would be smaller, then the old price is based on a wrong or outdated assumption about the company, and as that would get corrected, increasing the share price.
Should the share price increase by exactly 15 cents? No, most likely not, that would be technically possible but a rare coincidence. The share price reflects the expectation of all future profits, so if people expected that net profits this year would be 0 and it turned out to be 15 cents/share, the share price would likely grow much more than 15 cents/share because the expectation for the next years' profit would also probably increase.