Do your homework. The delinquent tax sale processes vary by state and from county to county. Generally, the state has rules/guidelines, but the counties can have variations in the implementation.
There are generally two types of tax sales. The first is a tax deed sale, where the winning bidder gets some type of limited warranty deed to the property which means you have work to do to "quiet title" and make sure no other lienholder interests are lurking around.
The other type of tax sale is involves tax certificates, where the bidder wins an interest-bearing certificate of sorts and must then initiate a foreclosure to gain title to the property. These are good because when the foreclosure completes (again check the states rules), you generally get title free and clear from other liens. However, that tax foreclosure requires time, attorneys, and fees.
Either way, you should expect to pay money at the tax sale, on the day of tax sale in most cases. Contact the real property tax office of the county you are interested in - they can provide more info on the process there. Oftentimes it is detailed on a website with plenty of FAQs.
It IS possible to buy a place for just the tax, but expect to make substantial repairs if you do get it cheap.
Regarding your last statement about how can a person OWN it, I'm not sure I understand...I can own my residence, and the bank can have a lien/interest for the mortgage, and my local municipality can have a lien/interest for property taxes, water bills, special assessments, etc. There can be judgments, liens, etc. out the wazoo on a property, which are independent of who holds title (OWNS) the property.