All icons and illustrations on Iconfinder are owned by the graphic designer who created them. This means that the designer is the owner of the copyright for the artwork. Iconfinder does not have copyright over the artwork.
When you are downloading icons or illustrations from Iconfinder, you are not getting the ownership or the copyright of them. You are only licensing them. Licensing means that you obtain the right to use the item under the terms of the license agreement.
All Premium content is protected by the Basic License. The Free items 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 item from Iconfinder, you have the right to use it forever, as long as you abide by the terms in the license agreement of that item. If you do not comply with the license terms, then the designer can stop you from using that item.
You can find information about the designer of an artwork by clicking the specific icon or illustration. 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 a designer, you can read more about ownership, copyright and licenses in this article: Copyright basics for icon designers