Copyleft is often presented as the opposite of copyright. We’ll see to which extent this can be true, but first let’s stress out that it isn’t 100% accurate (at least to me): in a sense, the total opposite of copyright is the public domain. Copyleft can be seen as a range of licenses providing a way of publishing free software while keeping some guarantees that we wouldn’t have with the public domain (without having to resort to a (proprietary) copyright license).

Non-copyleft licenses are the ones that are really opposed to copyright, and the closest to the public domain. Pieces of software like OpenSSH and Nginx are examples of software released under a BSD license.

The reason why other copyleft licenses are considered as the opposite of copyright, is that they use the rights granted by law to force people to make their work public and give up their rights (as they say: “all rights reversed”). In the case of strong copyleft licenses, the sole fact of using the software in another project forces the developer to release the whole project under a similar license. In other words, using such a license is like telling: “I worked for free, so if you use my work, then make it free as well”. The goal here is clearly to incentivize people to publish more software under free licenses, and avoid proprietary ones. This is also a way of deterring companies from making money off voluntary developers.
A big share of free software is licensed under the GNU GPL, for example Filezilla or Musescore.

While strong copyleft licenses are used for most free finished applications (ie. fully-featured desktop apps), it is less common for pieces of software that are used as bricks to build bigger software: libraries. In the case of libraries, weak copyleft is often preferred, allowing them to be used in both free and proprietary software. So, in short, the use of the software is unrestricted, and only modifications of the original software must be published with a similar license.
The main consequence the use of such licenses has, is that companies are no longer deterred from making money off free software. So the message is something along the lines of: “you can do whatever you want, but if you improve my work, make it available for others as well”, which is quite a different philosophy compared to strong copyleft.
As an example, the programming language OCaml is licensed under the GNU LGPL license. It can be also used for end-user software like Firefox, which is licensed under the Mozilla Public License, or 7-Zip which is also using the LGPL.

All in all, the choice of a license depends on the rights we want to give to (and the constraints we want to put on) people using and modifying our software: do we want them to always publish their work for the general good, or do we allow them to make a living developing upon it? It also depends on the type of software we’re writing (small libraries or big fully-featured projects?), as well as – of course – our ideological convictions.