There are several relationships being mentioned here. Also, this is pretty clearly taxes in the US, that's something important to specify since there are people on the site from around the globe.
When one company hires or contracts with someone to do a job -- either a contractor or a contracting business -- and the transaction is for more than $600, the company is supposed to file an annual tax form 1099-MISC with the IRS and send a copy to the company/person they contracted with.
The independent contractor or small business owner should receive a copy of the 1099-MISC. Even if the contractor/company does not get a copy, they should assume the money was reported to the IRS for tax purposes.
In all cases you need to make sure you pay your taxes as required by law. There are many free resources, contact your local SBA office or state tax office for free help, or a professional tax preparation company if you need help beyond that.
When it comes to annual taxes, there are at least two additional tax forms you probably need to use:
If you have your own small business or DBA, those funds should be reported on the schedule C ("Profit or Loss from Business"). This form gives you the opportunity to subtract the costs of doing business from your revenue, so you are only working on the difference. Schedule C has you enter all the income, but also lets you reduce that by the cost of goods, advertising, equipment, and so on.
If you earned funds that did not go through regular employment taxes you will need to file Schedule SE ("Self Employment Tax"). This form should be used for any income that didn't go through employment taxes. Lawn mowing, babysitting, neighborhood piano lessons, commissioned artwork, custom computer programs, whatever else you can think of. Most people use the profit/loss from your DBA business from Schedule C on the form rather than the raw income alone since it allows you to subtract your expenses.
For most of us who do side-business contract work it is typical to file both forms with personal income taxes.
Your state, county, city, or other local tax agencies may require additional forms. Check with your local tax professionals for more details.