When Exxon Mobil decides to buyback its own shares, it can decide to either go into the open market and purchase up to an approved limit of shares or it can present its shareholders with a tender offer (generally above current market price) for a direct share repurchase.
The bottom line of any buyback exercised by the company is that there will be fewer remaining outstanding shares. This translates into various benefits for investors: higher value per share, higher stake in the company and tax benefits due to lower capital gains. A stock buyback does not change the fundamental value of a company. Meaning a Exxon Mobil stock buyback will not make the business generate more revenue or be more valuable. It's simply that the value of the company is now shared between fewer shares. To further explore stock buybacks, please refer to Forbes.
If Exxon Mobil has decided to buyback its own stock, it's usually due to one of the following reasons: there is no better use for the cash on hand at the time, the price of shares has reached an unfair valuation and the company is betting on itself or it can also be to tame the new outstanding shares generated by stock compensation programs for employees.
Unfortunately, we do not know. There might be rumors of a Exxon Mobil stock buyback, but the truth is that until the board proposes and approves a share repurchase program, it's all just noise.
Fundamentally, the company is the same before and after a stock buyback has been completed. However, all per share metrics of the company will go up. EPS (Earnings-per-share) being the most notable and important metric that will go up by, roughly, the proportional amount of buyback being executed.
When you invest in Exxon Mobil, you do so because you are expecting to realize a profit from your investment. Depending on the company, this might be through one of two ways: share appreciation or dividends. Both are taxed differently and their tax codes can change throughout the years to favor one or the other.
Stock buybacks help return capital to investors through share appreciation. So, tax treatment would be the first difference. The second difference is that dividend programs tend to be periodic, meaning you can expect constant dividend payments every month/quarter/year. Historically speaking however, buybacks tend to be more of a sporadic way by companies to return capital to their shareholders when the financial conditions are suitable to do so.
If Exxon Mobil is performing a share repurchase program right now, it probably means they are in a good spot financially or at least, they are trying to pretend like they are.