Why are some programs in the Ubuntu Software Center not open source?

Several repositories

Ubuntu features various repositories with its own policies on the software being included there.

Summarized from Community Wiki - Repositories:

• main contains applications that are free software, can be freely redistributed and are fully supported by the Ubuntu team.

  This is the fully-FOSS section.

• restricted Our commitment is to only promote free software – or software available under a free licence. However, we make exceptions for a small set of tools and drivers that make it possible to install Ubuntu and its free applications on everyday hardware.

• universe is a snapshot of the free, open-source, and Linux world. It houses almost every piece of open-source software, all built from a range of public sources. Canonical does not provide a guarantee of regular security updates for software in the universe component

  Open source, maybe FOSS

• multiverse component contains software that is not free, which means the licensing requirements of this software do not meet the Ubuntu main component licence policy. [...] not supported and usually cannot be fixed or updated. Use it at your own risk.

These are the "regular" repositories which the Software Center uses to list the applications available.

Additional software (partners)

Additional to the above, there's the Extras and the Partner repositories. You'll have to enable those by yourself as an opt-in. The partner repository is more or less privately held by Canonical. It includes proprietary software in it, without source code available usually, just for reasons of being partners with Canonical or other commercial reasons.

Some costless "apps" in the Software Centre are provided via the Extras repository, and have to adhere to the Extension Repository Policy. These list just a few (see below).

The apps submitted via the MyApps programme are another story and very much Ubuntu specific. Source code is not always available and you can find some relevant information on the MyApps commercial software FAQ. Some are commercial and closed-source only, provided via a Private PPA with APT-HTTPS authentication. The ones distributed in this way don't have to adhere to any policies apart from the Launchpad PPA policies and some technical requirements - they're fairly private to the developers.

What about package x, y and z?

To see in which component a certain application is included, I suggest to go to packages.ubuntu.com, find the package, and in the page title you'll notice the section in brackets, e.g. Package: kcalc (4:4.9.2-0ubuntu1) [universe].

Can I run my system with only FOSS software?

To filter for FOSS-only, just only enable main (and optionally universe) - just to make sure you don't let in close-source software. Edit your /etc/apt/sources.list and files in /etc/apt/sources.list.d/ to exclude these if you like. You may run into issues running your hardware correctly as indicated by the description of the restricted component above.

Whether it will be very functional depends on your hardware and requirements you hold to the software you'd like to use. Some mainstream hardware doesn't require any proprietary firmware/binaries and can run perfectly fine without any closed source piece of data. The restricted and multiverse components provide a generally acceptable balance in this for most people, but you need turn them off in case you want to completely FOSS.

The set of applications you can pick will considerably scale down if you decide to disable the universe component. The list of all packages in Quantal has these numbers to give you some idea:

• main (+ security): 14250
• universe: 32595
• multiverse: 746
• restricted: 14

I was unable to find a credible source for the additional repositories, so by just using the regular Packages.gz files (assuming Quantal amd64):

• extras: 9
• partner: 9

See also

• Ubuntu Free software philosophy
• Extension Repository Policy
• How to remove the “For Purchase” section from the Software Center?

If you disable snaps using sudo apt purge gnome-software-plugin-snap as shown in another answer, and then in Ubuntu Software Center, go to Software & Updates > Ubuntu and uncheck the option that says "Software restricted by copyright or legal issues (multiverse)" I think that will remove all the non-FOSS results. At least, no proprietary results have shown up in my searches so far after doing so.

...except for drivers - uncheck the "restricted" option to get rid of those as well