I have Ubuntu 11.10 and want to upgrade to 12.04.5 LTS. I have Firefox 38 and Eclipse Luna installed from the official website. I also have many other apps installed, both from the repository and from websites in the Internet. So, if I upgrade, will any of my apps get deleted?

Official packages of software that are available in one branch are likely available in the next branch. The exception is if they have been removed due to being unmaintained or too old to be compatible with newer software technology. This is the case for any operating system ever created, at any point in time, that a computer has been utilized.

What you can do is check the release branch repositories you are upgrading to and find out for yourself if the software is still available. If it is not, build a back ported package for your distribution.

Edit:

I failed to mention that unofficial packages will probably be marked as obsolete by apt, aptitude and synaptics in the case of a system upgrade. It has been quite some time since I ran Ubuntu, but usually the software that is not upgraded will remain installed, unless it is interfering with another package. Your package manager should ask you what you want to do with it if it is not compatible with some part of your system. Sometimes obsolete software remains functional, sometimes it breaks in this type of situation. You will have to experiment with it and see what happens.

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Thank you for your kind help. I have never upgraded a OS before.

this might cause problems.
you should always prefer installing from the repositories.