Your first step is to explore what it takes to formally “start” a business, and which of those items cost money.

Planning.

You’ll need to come up with a business plan and financial model, of course, but you can do this on your own, for free.
Business license.

If you’re planning on creating a partnership, LLC or corporation, you’ll need to file some paperwork — but it probably won’t cost you more than a few hundred dollars, depending on what licensing you need. The Small Business Administration has plenty of resources to help you figure out what you need, how to obtain it and how much it will cost.
A domain name.

You’ll need to invest in your online brand early on; while I suggest going as professional as possible, you could also use a bare-bones approach to launch, if yours is a minimum viable product. Often, a catchy domain name is all you need to define your brand at the start, and one can be bought for as little as $10 (if you can find one that isn’t taken!). I use GoDaddy to buy domains.
A website.

Website builders these days are free and intuitive to use. You won’t expend anything but time to build your first site. I recommend starting simple with a widely-used website platform, like WordPress.
Marketing.

While marketing has a reputation for being very expensive, there are actually a ton of really effective tactics that can be performed with only an investment of your time. Social media marketing, SEO and content marketing all fit within this category — and, honestly, those are really all you need. For help, see The Definitive Guide to Marketing Your Business Online.
Equipment.

Equipment, offices and other tangible assets are cash killers, but not all businesses need them. Some businesses don’t require any of these things, as I’ll explain shortly.
Products. Finally, all businesses need to sell something, which usually means some up-front investing. However, many services can be performed with an investment of time rather than money.
So, which types of businesses can be started without a heavy financial burden in any of the above areas?