All icons on Iconfinder are owned by the icon designer who created them. This means the icon designer is the owner of the copyright for the icon. Iconfinder does not have copyright over the icons.
When you are downloading icons from Iconfinder, you are not getting the ownership or the copyright of the icons. You are only licensing the icons. Licensing means that you obtain the right to use the icon, under the terms of the license agreement.
All Premium icons are protected by the Basic License. The Free icons may be protected by different licenses, so you will need to pay attention to each license. A good overview of most important licenses available on Iconfinder can be accessed in this License overview table. Check out the License overview article for more information.
Once you download an icon from Iconfinder, you have the right to use the icon forever, as long as you abide by the terms in the license agreement of that icon. If you do not comply with the license terms, then the icon designer can stop you from using that icon.
You can find information about the designer of an icon by clicking the specific icon. The name of the author and a link to their Iconfinder page can be found in the top right corner, as in the screenshot below. You can contact the designer directly on Iconfinder by sending them a message.
If you are an icon designer, you can read more about ownership, copyright and licenses in this article: Copyright basics for icon designers