The pH meter & the pH electrode
pH meter is a device for pH measurements.
pH meter is nothing else but a precise voltameter (see example pH meter circuit diagram here) connected to the pH electrode - kind of ion selective electrode. Voltage produced by the pH electrode is proportional to logartithm of the H+ activity (as described by the Nernst equation). pH meter voltameter display is scaled in such a way that the displayed result of measurement is just the pH of the solution.
Commercial pH electrodes used in pH meters consist of a H+ selective membrane (which can be made made just of very thin glass), and internal and external reference electrodes, usually combined in one housing.
Although there are some restrictions on the use of the electrodes and the way they are treated between measurements, pH meters are in most cases the best way to check pH of the solution, as they are much more precise than indicators and pH stripes (papers). Using properly calibrated pH meter with a good electrode one may measure pH with +/- 0.01 unit accuracy without any problem. In carefully prepared experiments it is possible to obtain much better precision of measurement.
Note, that pH meter measures not the H+ concentration, but activity, as all other ion selective electrodes do.
It is also worth of noting here that - while pH meter allows very precise measurement - in many case approximate pH value is precise enough. In such cases, instead of using pH meter, it can be much more convenient (and much cheaper) to use simple pH test strips for acid-alkaline measurements.
You will find much more information about pH meters and pH electrodes on the specialized www.ph-meter.info site.