Open data refers to data sources that are freely available for reuse by others with no or minimal restrictions. Typically, the only condition sometimes found on an open dataset is a requirement to cite the original data source. International and government agencies are some of the most common organized sources of open data, but many researchers choose to release specialized datasets openly as well.
Keep in mind that open datasets, like all sources of information, should be evaluated for their quality.
Due to new data sharing requirements by publishers and federal funders, an increasing amount of research data is being publicly shared and made discoverable. To determine how you are permitted to reuse a dataset, check to see if it has been assigned a copyright license. Datasets that are indicated as being in the public domain (sometimes expressed as CC0) are free to copy and reuse without restriction. Some datasets are shared under Creative Commons licenses, which are meant to provide a simplified license to promote reuse. For example, a CC-BY license indicates that you are free to copy or reuse a dataset (or other resource) without restriction, provided that you give attribution to the original creator.
The US government maintains a data catalog with more than 250,000 datasets from government organizations at every level. From the data catalog, you can filter by level of government, topic, data format, and more.
These links were pulled from the list "Open Data in the United States" made available at data.gov.
For research data, you can find discipline-specific datasets (many of which are openly accessible) using the Registry of Research Data Repositories.
For interesting datasets across a range of areas, subscribe to the Data Is Plural newsletter (or view its archive of previously shared datasets).