Really big birds are an interesting category in cryptozoology.  Most people consider these to be nothing but fictional legends, the simple up-scaling of existing large birds like bald eagles and condors.  It certainly doesn't take much imagination to invent "something just like that bird over there, only bigger."

However, there are also vexing reports of really big birds which occasionally surface every few years.  Cryptozoologists speculate that some of these reports could be sightings of surviving pterodactyls (which went extinct X million years ago), or of teratorns (which went extinct only 6,000 years ago).  Teratorns were larger relatives of the condor, and lived in North America.  Many excellent examples of teratorns have been recovered from the La Brea tar pits in Los Angeles.

One problem with thunderbird sightings is that they rely on A) correctly identifying a bird as "unknown," and B) correctly identifying the scale of something in the sky.  The ability of an untrained observer to perform both of these feats at once is unlikely, to say the least.  (Witness all the reports of various mysterious wild beasts which turn out simply to be house cats crossing a distant field.)  Most thunderbird reports are probably simply a case of mistaken identity, and can be attributed to eagles or great blue herons.

The Thunderbird is a sacred animal of Native American legend, representing power and strength.  Thunder is said to be the sound of the thunderbird's beating wings, and the bird itself is spotted riding before storms, perhaps because its large size can be born aloft on the thermal updrafts which accompany thunderstorms.

The Roc is the Old World version of the thunderbird.  Rocs derive from ancient Arabic and Persian legends.  They were said to subsist primarily on elephants, which they carried away to their nests.  The roc shares a mythic heritage with the phoenix, which can be considered a "related animal."  Although the phoenix is smaller, and is most noted for its habit of combusting and being reborn from its own ashes.

There is a sub-mystery to the thunderbird story: countless people remember having seen an antique photograph of a thunderbird nailed to the wall of a barn, with several men standing around it in old-timey period clothing.  However, it seems that this photograph never existed.

Several possibilities have been floated.  One is that we are all mis-remembering another photograph - possible candidates include an illustration plate in Arthur Conan Doyle's "The Lost World," an illustration used by Ripley's Believe It Or Not, and a photograph of men holding an Andean condor which may have appeared in the Guinness Book of World Records (or a similar tome).  Another possibility is that the specificity of the description essentially creates a false memory in the reader.  "Oh yeah, I remember seeing that!"

Quite a lot of demonstrably false pictures of thunderbirds have been uncovered. All have been debunked as fairly recent forgeries, notably a "Civil War era" picture, which was used in the short-lived television show Freaky Links.