Analog signals : Analog signals are continuous signals. Analog electronics deals with analog circuits. Analog circuits are circuits which process or work with analog signals or continuous signals.
Digital signals: Digital signals are not continuous signals. Digital signals are discrete pulses or signals at certain time intervals. Digital electronics deals with the study and design of digital circuits. Digital circuits are circuits which process digital signals or discrete signals.
If we continuously measure the temperature of a room and plot it for every instants of time, then the graph will be continuous and we can call it an analog signal. The same temperature if we measure it at certain time intervals, say each hour gap between two readings and if we plot the temperature, the graph will be having discrete readings or pulses at each hour. This signal is now called discrete signal.
Another example for analog signal is sound signal.