Bit rate and data rate are somewhat ambiguous, and their exact definitions vary from one field or application to another.
Bit rate is often used to measure the information rate, since information is measured in bits. Sometimes it is used to measure the number of actual 1s and 0s transmitted per second. When the information is compressed (source-coded), the two definitions are equal.
Data rate can be used to mean the same as bit rate. Sometimes it is used to measure the rate at which data, as opposed to overhead, is transmitted. For instance, an audio compact disc system has a data rate of $2\cdot44100\cdot16=1411200$ bits (zeros and ones) per second, but the actual transmission rate is approximately double that, once you include framing, metadata, and FEC.
The baud rate (or symbol rate, or pulse rate) is the number of actual pulses (symbols) transmitted per second. If a pulse "carries" several bits, then the baud rate can be significantly smaller than the bit rate. This is important because the bandwidth required to transmit the information is determined by the baud rate.
So, consider an encoded video stream with 1000 bits per frame, at 24 frames per second, with 10% overhead per frame, and 4 bits per symbol:
- From an information-theory perspective, the bit rate is $24\cdot 1000=24000$ (information) bits/sec.
- You can also say, from a more practical perspective, that the bit rate required is $1.1\cdot 24 \cdot 1000 = 26400$ (ones and zeros) bits per second, but you could still define the data rate as $24000$ bits per second, since the remaining 2400 bits are overhead.
- The pulse rate on the transmission link will be $26400/4=6,600$ pulses per second.