An operational amplifier (op amp) is an integrated circuit device. Nearly all synth modules use op amps. They can be used to amplify (up or down) signals, add or subtract signals, buffer signals, determine if a signal is above or below some threshold, drive LEDs, provide a current source… they do a lot.
A single chip may be a single, dual, or quad op amp, having 1, 2, or 4 individual op amps inside it. There are many op amps, having different specs, on the market. In synth designs the ones you see most often are TL07x (where x is 1, 2, or 4, meaning single, dual, or quad). MCP600x are also fairly common.
The symbol for an op amp in a schematic diagram is a triangle:
The different op amps in a dual or quad package will be shown as separate triangles, and the power connection may be shown separately:
An operational transconductance amplifier (OTA) is similar but provides a current, rather than voltage, output, and its gain can be controlled by applying a control current. OTAs are often used in voltage controlled filters and amplifiers. By far the most common (dual) OTA in synth designs is the LM13700. Besides the two OTAs, it contains two transistor pairs for buffering.
