How are homebrew programs done?

They're written in some type of Computer language (not sure, I'm no expert whatsoever).

You can try FatQuack's DS Game Maker 2.3 for making a DS game...

 

You got to know how to code, and things about the ARM structure.
i don't know the details, but it aint easy.
Anyway.. there already are Calculator homebrews ( I.E it is implemented in DSOrganize)

 

mainly they are done usng c++ but there are some programs that you can use

 

mainly they are done usng c++ but there are some programs that you can use

 

There are a bunch of homebrew threads but 3 things

high level: there is a very limited java implementation, lua (DSlua), basic and the like and I will lump in the game making/scripting stuff. I am probably missing a few but those are the main ones I can think of.

not quite as high level: this is your C/C++ and is the workhorse of DS programming. C is a bit faster and can make some tighter code. Notable toolkits include palib and devkitpro. The latter is a group of libraries to sort the hardware and a compiler/assembler kit
Palib on the other hand is quite high level. It has somewhat of a reputation for being a bit clunky and slow but some damn nice homebrew has come from it and you can still backport as it were to the devkitpro toolchain when you grow out of it.

ASM level: the DS has 2 arm processors. An ARM9 and 66MHz and an ARM7 at 33MHz. This is mainly used for stuff like emulation, audio and video and hacking commercial roms as it can make the tightest code. Inline assembler (where you write a small section in ASM to do a particular task) is possible too. There are a couple of assemblers available but can I suggest the one with devkitpro, it is not quite as nice as some of the commercial stuff but when commercial means several thousand....

DS hardware specs (whatever you end up with I suggest you read these):
http://nocash.emubase.de/gbatek.htm#dstechnicaldata

As for calculator the ti85 and similar have fairly good emulators and there are a bunch of calculators with more simple functions.

 

Here is a very good tutorial for using palib.

http://www.palib.info/wiki/doku.php

It also explanes how to set everything up.
You should be familiar with c to understand whats going on.