Two kinds of cheap board show up all over ham shacks. An Arduino-class microcontroller runs a single program and handles pins and timing. A Raspberry Pi-class single-board computer runs a full operating system such as Linux, with networking, USB and a screen. Both are small, inexpensive and easy to start with, and picking the right one is most of the design.
Which one for which job
Rule of thumb: timing and pins, microcontroller; software and networking, computer. Typical examples only.
Ask what the job needs. If it is timing and pins (generate a dit exactly 60 ms long, read a potentiometer, switch a relay) a microcontroller does it with less power, less cost and no operating system to boot. If it is software and networking (decode FT8, run a web server, talk to an internet gateway) a small computer is the better fit. Many projects use one of each: a Pi for the software, a microcontroller for the real-time job.
A tiny example of the kind of arithmetic a keyer does, using the Morse unit of 1.2 ÷ WPM seconds (see The Morse alphabet and timing):
// 60 ms at 20 WPM
int dit_ms = 1200 / wpm;
What hams build
Digital-voice hotspots. A Pi links a handheld to internet-based digital voice networks; see Hotspots and gateways.
Headless digital stations. A Pi runs software such as WSJT-X (FT8 and FT4) with a USB audio interface and rig control. Pair it with a remote link and you have a remote station.
APRS trackers, digipeaters and gateways; see APRS.
SDR hosts, loggers and clocks. A Pi can run an SDR dongle and share its output over the network, keep a logbook, or serve accurate time from GPS.
Doing it safely
Isolate the radio. Key a transmitter through a transistor or optocoupler, and use a USB audio interface with isolation. A direct wire from a board to a radio invites hum, damage and ground loops; see Computer and sound-card interfaces and CAT control and rig software.
Respect the logic level. A Pi's pins are 3.3 V only. 5 V on one can destroy the board.
Keep RF out and noise down. A transmitter nearby can crash or confuse a board, and switching power supplies, USB and clocks can spray noise into your receiver. Use a metal case, ferrites and short leads; see Common-mode current and ground loops.
Shut a Pi down cleanly. Cutting power can corrupt the storage card. Use a proper shutdown, or a design meant for sudden power loss.
An unattended station is still a station. A hotspot, beacon or gateway must be under proper control and identify as the rules require; see Control operators and control points.