No one is sure exactly how many galaxies there are in total—many millions for sure! To illustrate how many galaxies there are that are known, a single 2 year survey (the 2dF Galaxy Survey) which finished in 2003 surveyed 250,000 galaxies in order to make a 3D map of the Universe. And that's certainly not all known galaxies! Check out the 2dF website for more on this. As another example, there are 357 million objects in the 7th data release of the Sloan Digital Sky Survey (SDSS), although many of those are stars, and SDSS has specta for about one milion galaxies. There are about 215 million distinct known galaxies in the NASA Extragalactic Database (NED)! And 515 million sources which still need to be anlyzed to determine if they're new objects or just new measurements of known old objects. This is still probably not everything either. When a new telescope comes online that has a significant upgrade in observing power compared to what was available before, Astronomers will generally do a big survey to find new galaxies, and it takes time for these new galaxy catalogs to be merged with big databases like NED. So the exact number of known galaxies is constantly changing!
For the observable Universe, it is estimated that there are as many as 200 billion galaxies, but we aren't able to see all of them as our telescopes are not sensitive enough. In addition, different types of telescopes are better at finding different types of galaxies. For example, many very distant galaxies (galaxies early in cosmic history) are very dusty, and dust blocks the optical starlight. Therefore optical telescopes like Hubble won't see these dusty galaxies. However, that dust gets hot from absorbing the starlight and then emits light in the infrared, which can be detected with infrared telescopes like Herschel. Collating these different galaxy surveys together can be challenging and affects estimates about the number of galaxies in the Universe.
This page was last updated by Chelsea Sharon on July 18, 2015