We have taken an open-source application, released under GPL ver1.09 and have modified some parts of its source and have made some customisations/enhancements as per our requirement.
We are using this application for non-commercial, internal purposes only.
Is it mandatory that we have to publish the modified code in open-source, if we wish to use it?
EDIT : Does it vary in any way, depending upon its user-base? (e.g. Just the company employees or outsiders as well?)
If you're not going to distribute it, you can do what you want with it.
Adding to Jon's answer, if you distribute it in any way, the GPL license is deliberately viral, so you'd have to provide source to the entire application with the distribution.
Short answer: if it's GPL, you have to:
As others have stated, you will only need to give the source to people you have given the binary to (and then only if they ask for it). So, if the binary is only used within the company, you don't need to hand the soure out to anyone but employees (if they ask for the source).
You cannot, however, restrict someone from passing the source (or the application) on, but I cannot answer if any company confidentialy agreement would trump or be trumped by the GPL.