It is very difficult to give a reasonable estimate for the total number of stars and galaxies in the Universe, because we cannot always draw a well-defined line between stars and non-stars, or between different galaxies.
A star is usually defined as a ball of gas that is kept together by its own gravity and in which nuclear reactions take place or took place. Whether nuclear reactions are happening in a ball of gass depends mostly on the mass of the gas. A ball of gas can only become a star if it has enough mass. According to our theories, the dividing line is at a mass of about a tenth of that of the Sun. If a ball of gas has at least that much mass, then it can become a star. If the ball of gas has less mass, then it is a so-called brown dwarf, or a planet (such as Jupiter).
There are three problems with this definition of a star: Firstly, you cannot always tell from the outside whether nuclear reactions are happening on the inside, and how many. Secondly, nuclear reactions don't suddenly turn on at a particular mass, and at slightly smaller masses there are also already some nuclear reactions. Thirdly, there are very many more balls of gas with small masses than with large masses, so for the exact number of stars it is very important at what mass exactly you draw the line.
A galaxy is a collection of gas and dust and stars that orbit around a common center of mass and that are separated form other galaxies by vast expanses of empty space, but this definition is not sufficient, because it doesn't say how large or empty those expanses must be. Cases are known of multiple galaxies being in the process of coalescing into one, and then it is not clear when they should still be counted as separate galaxies, and when as just a single one. It is also not clear how small such a collection of gas and dust and stars may be to yet be called a galaxy. A star that escaped from a galaxy a long time ago and that now drifts alone through space between galaxies will probably not be called a galaxy by anyone, but perhaps a thousand stars orbiting around each other might be. And just as for balls of gas, there are very many more small galaxies than large ones, so for the precise count it is very important at what size exactly you draw the line.
All in all you can get very different answers, depending on the lower boundaries that you put on balls of gas and on collections of stars.
It is often said that a typical galaxy contains on the order of 100 thousand million stars, and that the number of galaxies in the visible Universe is about that same number. With these assumptions, the number of stars in the visible Universe would be about 1 × 1022. However, it is not clear which galaxy is typical, and whether the typical galaxy that contains 100 thousand million stars is also the typical galaxy of which there are 100 thousand million. According to http://www.space.com/scienceastronomy/star_count_030722.html there are about 7 × 1022 stars in the visible Univese that could be observed by our telescopes in principle, but that number probably does not include very many dim stars.
If my calculations are right, then the SKY-model of our own Milky Way Galaxy (Wainscoat et al, Astrophysical Journal Supplement Series, volume 83, pagina 111) says that in the disk of our Galaxy alone there are already about 2.2 million million (2.2 × 1012) stars, of which about a quarter are very dim dwarf stars of spectral type M (spectral classes greater than or equal to M5), about half are less dim M-dwarfs (spectral classes M0-4), and a quarter are brighter stars (earlier than M or brighter than V). This total count is about a factor of 20 greater than the typical value that is often mentioned, even though our Galaxy is often seen as fairly typical.
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