"Closed software/source" is when software is created that is private and nobody can look at the source code. Traditionally speaking, software companies usually create closed source code to stop competitors from learning their tricks and making similar systems or getting any ideas.
"Open software/source" is when software is created and anybody can download the source code to either use or change it (according to licenses...). Many of the large internet companies around today pretty much run on open source; the majority of the world's web servers (essentially, the internet) run on open source Linux. It's particularly important for security, too. For example, iMessage is supposedly point-to-point encryption but we have to take Apple's word for this. If Apple made the source code open, all of the security researchers would quickly verify this.
There are pros and cons to each but hopefully that explains it.