There is no unique definition, but in general baseband signals have a spectrum that starts close to DC. Most analog information sources that are considered in communications are baseband: audio, video, signals from sensors, images, etc. Also, any signal that is a sum of pulses of different shapes is a baseband signal; examples are NRZ and Manchester-encoded signals. Yet another definition is that a baseband signal has not been modulated, or its spectrum has not been shifted, for the purpose of matching it to a particular communication channel.
Passband signals were originally baseband, but were modulated, or frequency-shifted (or up-converted) for some purpose, usually to make them fit in a passband channel. Consider the case of AM radio. An audio signal is low-pass filtered to 10 kHz, resulting in a baseband signal with a spectrum from 0 Hz (or strictly, 20 Hz) to 10 kHz. Then, this signal is shifted up in frequency (upconverted) to make it fit into the 20 kHz channel assigned to the AM station. The receiver down-converts the signal back to baseband and plays it over the loudspeakers.
As you say, the upconversion process involves changing the amplitude, frequency and/or phase of a high-frequency carrier signal proportionally to the information-bearing baseband signal. The receiver uses the amplitude, frequency and/or phase of the carrier to reconstruct the baseband signal.