Magnetic islands formed after magnetic reconnection I often read about magnetic islands forming as a result of magnetic reconnection? 
To my understanding, those are irregularities of the magnetic field right? If not, could someone help me understand this concept?
 A: Magnetic islands in magnetic reconnection can be a few things, depending on the dimensions of the simulation and geometry of the region in question.  In a 2D simulation, magnetic islands are genuine islands in the sense that they are isolated regions of magnetic flux.  That is, the magnetic field lines bounding them are all closed.  In principle, a particle with a small enough gyroradius could orbit the center of one of these islands by following the closed magnetic field lines.
The islands form when there are multiple reconnection sites allowing for a region of closed magnetic flux to form.  There is a nice short letter on this by Jim Drake et al. at http://terpconnect.umd.edu/~drake/publications/reconnection/drake06.pdf with a few figurative examples.  If you have journal access, there are more examples and explanations in the following article: https://agupubs.onlinelibrary.wiley.com/doi/abstract/10.1002/2013JA019483

To my understanding, those are irregularities of the magnetic field right?

Well, kind of but I am reluctant to call them irregular since they seem to be a regular feature in regions of turbulent reconnection.  They are irregular in that they tend not to exist in isolation.
