Streams were introduced with PHP 4.3.0 as a way of generalizing file, network, data compression, and other operations which share a common set of functions and uses. In its simplest definition, a stream is a resource object which exhibits streamable behavior. That is, it can be read from or written to in a linear fashion, and may be able to fseek() to an arbitrary locations within the stream.
A wrapper is additional code which tells the stream how to handle specific protocols/encodings. For example, the http wrapper knows how to translate a URL into an HTTP/1.0 request for a file on a remote server. There are many wrappers built into PHP by default (See List of Supported Protocols/Wrappers), and additional, custom wrappers may be added either within a PHP script using stream_wrapper_register(), or directly from an extension using the API Reference in Working with streams. Because any variety of wrapper may be added to PHP, there is no set limit on what can be done with them. To access the list of currently registered wrappers, use stream_get_wrappers().
A stream is referenced as: scheme://target
Information on using streams within the PHP source code can be found in the Streams API for PHP Extension Authors reference.