How is a star's parent galaxy recognized? A star is probably visible/detected by it's radiation. But that star may or may not belong to our own galaxy ... yet news reports speak of detecting a star/nova in a distant galaxy. 
How does one determine whether the star she/he views belongs to Milky Way, or some other galaxy ... or is galactic orphan? Is it merely a matter of the distance to that star?
 A: Every ordinary star we are able to individually observe is a part of the Milky Way. Well, except for stars in a small number of very nearby galaxies but even galaxies such as Andromeda look like a "continuum" so we're not observing the stars individually although we see that the galaxy isn't just a point.
Only if a star goes nova (a lethal nuclear explosion of a white dwarf star) of supernova (a similar explosion but stronger), it may be observed outside the Milky Way. In all such cases we've experienced, one may always identify a galaxy at the same location that was known before the nova/supernova explosion. So the star going nova/supernova clearly belongs to that galaxy. Please note that distant galaxies look like dots – pretty much visually indistinguishable from stars in the Milky Way.
A star going nova has 50,000-100,000 times higher luminosity than the Sun; the number is even higher for a supernova. That's a sufficient increase of the luminosity for an exploding star in a distant galaxy to become "almost as bright" as the whole galaxy, well, not quite.
A: The key is, that the intrinsic brightness of all supernovae (at least for the most important type 1a) is roughly the same: it peaks around magnitude -19. From the difference with the apparent magnitude (called distance modulus) the distance to the supernova can be derived, and then compared to the distance to the suspected host galaxy (derived from its redshift).
The problem is that for both distances, a model of universe must be assumed. But that is another question.
For nearby supernovae and novae, the size of the expanding photosphere can be used as an alternate method for distance estimates too. This case is less usual.
With some quasars the same question arises, because there is a galaxy in the line of sight. In this case, the difference in redshifts, and thus in distance, leaves no doubts, although both objects appear at the same position in the sky.
A: The stars of our own galaxy are always much brighter than the stars of other galaxies. Just as a point of reference, the Milky way is about 100 thousand light years across. The nearest large galaxy, the Andromeda galaxy, is 2.2 million light years away. All of its stars would therefore be 20 times as far away than any star in our galaxy. 
Here's a picture of the Andromeda galaxy, for example:

(source: solstation.com) 
The stars that belong to the Andromeda galaxy are mostly not even recognizable as stars at all. They make up the fuzz of the disk. There are also two satellite galaxies that appear fuzzy in the image. Most other points of light are stars from our own galaxy, with a few faint fuzzies in the very distant background that are other, far more distant galaxies. 
Every other galaxy (and there are billions) is many times farther away, with the exception of the large and small magellanic clouds, which are satellites of the Milky Way.
This coincidentally, was the reason why up until Edwin Hubble in the early 1900s, noone was able to determine whether galaxies were nebulae or separate, distinct objects from our own galaxy. It wasn't until the construction of the 100 inch Hooker telescope and the later 200 inch Mount Palomar telescope, that anyone was able to resolve any stars at all from these galaxies.
