Accidental Formatting
Formatting can mean very different things. A quick format mainly replaces file-system metadata and may leave much of the old file content recoverable; a full format or secure erase can overwrite or discard data and needs to be treated as a different case.
What's actually happening
A quick format creates new file-system structures without deliberately overwriting every data block. That can replace important metadata while leaving much of the old file content elsewhere on the device. A full format is not equivalent to a quick format: for example, modern Windows full-format operations write across the formatted volume while checking it, so old file content in that area should generally be considered overwritten. Secure erase, sanitize, and device-level erase operations are different again and may make recovery impossible.
How recoverable this really is
After a quick format, recovery can be good if little or nothing has been written afterward, but results depend on the file system, the type of storage, and exactly what the formatting operation changed. New file-system metadata may overlap old metadata, so original names, directory structure, timestamps, and fragmentation information can be damaged even when some file content remains intact. On SSDs, TRIM and controller behavior can reduce recoverability further.
The general approach
If a backup copy of the old file system's index partially survived the format — which happens more often than people expect, especially with a quick format that reused the same file system type — recovery software can sometimes rebuild the original folder structure and names directly from it. Where that's gone, the fallback is the same signature-based carving used for plain deletion: scanning the raw disk for recognizable file headers, which recovers content reliably but generally without the original names.
Don't format the drive again "to see if that fixes it," and don't save anything new to it while deciding what to do next. A second format is one of the more common ways this situation goes from recoverable to not, because it gives the new file system's metadata a second chance to land on top of whatever survived the first time.
After a quick format on a healthy device, file-system-aware recovery software can scan the old volume and, where metadata survives, reconstruct more than signature-based carving alone. See the directory of recovery software by vendor for available tools. If the formatted device is an SSD, also read SSD deletion and TRIM before assuming the same recovery expectations as a hard drive.
This page describes the general situation, not a guaranteed fix for your specific case. If you'd rather have someone experienced take a look before you try anything, see what a hands-on evaluation looks like.