There are two points of overlap between your employer and your side business to be mindful of.
1) time and resources.
Generally speaking, your employer owns all IP you develop at work, or using work resources. In practice, I've never seen an employer be a dick about this, but you should still be exceedingly careful. It's really easy to be 100% free and clear on this one. Just don't ever do side project work (to include answering emails, etc) on work machines or on premises at work.
2) intellectually property.
This one's trickier. The agreement that matters most you already signed, if your employer uses one. It's basically a list of stuff you've already invented or have as a side project, and it exists specifically as a papertrail. eg: I work on finance software interfaces as my full time job, and have also done work on this for myself on the side. This agreement says "I created product X, which does Y with method Z" as part of the employment agreement, so that they can't later claim that I only came up with Y and Z because of my employer's training or resources, thus entitling employer to the project.
How careful you need to be here depends on the thematic proximity of your employer and your side project. If you work for dropbox and your side project is flappy bird, you're probably fine. If you work for adobe and your side project is an alternative to photoshop, do a lot of homework and consider talking to a lawyer.
TL;DR: don't use employer time or resources, do use common sense.
Those may be illegal in your jurisdiction but in most places in the US they arent.
You should verify this before starting to look for part time work.
Respectfully, I'd argue that building your own skill set or business is going to be exponentially more valuable use for those hours anyway, unless you really need the cash.
I've been seeing a lot of arguments that instead say that 6-hour workdays would make employees more productive. Well...
I work about 20hrs a week on various side-gigs. Pretty strict about the working hours and work not overlapping with whatever my current employer is doing. It's nice to keep things separate.
Obviously, that only applies if you do actually turn up tired all the time. If you can pull off the longer hours, rock on!
In my contract, there is a clause that has an issue with this, but only if a conflict of interest occurs. Since both companies are in very different industries (defence & mobile), this hasn't been a problem so far.
I don't remember whether this is stated in my employment agreement or not, but practically freelancing is OK, as long as you get the job done first. Do it after office hour, for example.
I'm in Indonesia, BTW.
For example 'we need a tweet this button in the side bar below the search box'.