Serial, RS232, Modbus, UART and TCP
Serial is a general communications scheme, where information (in this case binary data) is transmitted in pieces sequentially. There are a number of compatible and incompatible computer buses and protocols that use a serial communications.

In the context of your question, serial is used as both a communication scheme and a description of the physical bus. That is, the chips on either end of the cable/wire use a sequential sequence of bits to exchange data. Commonly "serial" is used to refer to a number of buses that were generally available on PC/AT and compatibles via the "COM" port, generally RS232 and RS485 are included.

Confusingly both the physical interface and the protocol could use a serial communication scheme, or they could be different (e.g. serial data over a parallel link that shifts a byte at a time instead of a bit) so its important to distinguish between the two. A general picture of this difference is given in the OSI model, where every layer can use whatever communication scheme they choose independent of each other



RS232 is a standard that defines a physical communication scheme using a serial connection. Similarly RS485 does the same. Both of these standards dictate the physical serial bus (with voltage levels and timing details) behind the communication scheme (that could be ModBus or a proprietary printer peripheral comm protocol)

Modbus is a standard that only defines a common high level (OSI 7) protocol but leaves the physical details undefined (implementation). Officially can be implemented over TCP/IP or over a serial bus, Modbus is high level protocol and TCP/IP (for example) can be implemented even over carrier pigeon