How is sea level measured?
There are two main approaches to measuring sea level (and thus its change): local and satellite-based.

To measure local sea level we can set up gauges on the coast that record the water level every so often - hourly is common. These use various methods, which have evolved over time, all aimed at having a very precise (sometimes down to millimetres) long-term record at that location. In the UK, the UK Tide Gauge network has records going back (in some locations) to 1915.

The mechanisms of tide gauges vary, and have improved over time. Early ones used floats inside a "stilling well" (a column of water that protects the gauge from waves). Some modern gauges work by sensing water pressure. If we have good measurements of temperature and salinity we can calculate density, and if we know density and the pressure at a location, we can calculate the amount of water above that location. Other modern gauges use ultrasound or radar transceivers located at a known vertical position above the water, and thus measure the distance to the water surface with very high accuracy. 

These records of water level include the tides, of course, as well as other influences such as the effects of changing air pressure, wind driven currents, and so forth. They are also affected by the rising and falling of land over time. These effects can all be accounted for, and removed from the signal, and effort goes into doing this. You might some interesting reading on the topic from the National Tidal and Sea Level Facility.

However, no matter how good the record from fixed tide gauges, they can only tell us what is happening on the coast. While that's arguably the most important for mankind, it's hardly a global picture.

The other approach that we can take is satellite observations. Radar altimeters on satellites can, and do, measure sea level worldwide to a precision of a few centimetres or better. This is a recent mission for this purpose (there may have been others since - I'm not an expert). The level of accuracy here is not as good, but it can be improved statistically through a large number of measurements. 

The greater difficulty with satellite altimetry is removing the tides, because the satellites are always moving over the earth's surface and so can only measure any one location at relatively infrequent intervals. Consequently, some sophisticated global tidal models are used with data provided by the satellites to (a) improve the models, for people who want to predict tides, and (b) use the tides from the models to remove the tides from the satellite measurements, leaving just longer-term changes in sea level.

I think part of the intent of your question is "people quote changes in sea level of millimetres or centimetres per year, yet surely the measurements are not this accurate?". Well, satellite measurements aren't, though local ones might be, but remember that we're measuring for more than just one year. One approach might be to look at a ten- or twenty-year trend, and work out the average change per year from that.