Digital audio formats are a bunch of numbers. A sequence of those numbers transported to somewhere else is therefore a stream of bits ('bitstream'). The bits are transported one after another, like a single file line.
There are raw formats like PCM where each number represents the amplitude of the original analog signal at various points in time (eg. 16 bit / 44.1 kHz PCM on a CD means there are 44,100 16 bit numbers for every second of the audio; that means to digitize the original analog signal it was 'examined' 44,100 times per second).
Other formats compress the raw data into another form to save space or to include more discrete channels of information in the same available space. Those are still transported one bit at a time but need additional processing ('decoding') to turn it back into the raw PCM. So a DD/DTS or other bitstream transported to your receiver requires the receiver to decode it to a raw sequence of numbers which can then be converted to analog and amplified.
S/PDIF is data format and communication protocol for transporting digital audio from a sender to a receiver (or source to sink, if you will). If you are using coax or optical digital connections from say a dvd player to an AV receiver you are using s/pdif. HDMI is the same principle but can carry audio, video, control signals, copy protection data, etc all at the same time.
Re computer sound cards: There are some cards that include the ability to output a digital signal only so that you could send it to a receiver for decoding 5.1 but many simply include the processor, either in the hardware or as part of the software (driver), to do the decoding and output each channel as an analog signal after decoding.