Types of computation are not based on the type of device but on the way in which information is structured and processed. The principal types are analogue, digital and quantum. (What follows is updated following comments and downvotes made).
For example an abacus is a digital (ON/OFF) computer, albeit using decimal word lengths rather than bit-count word lengths. Some early electronic computers used the same decimal system, as did Babbage's mechanical Difference Engine of the late 19th century. A box of counters, a pencil and a sheet of paper are an even simpler, though rather more flexible, digital computer. Powerful ones just happen to be electronic these days.
By contrast an analogue computer processes signals (numbers) which are continuously variable across the working range. Analogue computers have been mechanical as in the computing gunsights fitted to American bombers in WWII, electrical as in some early aircraft control systems, or even hydraulic as in the first ever computational model of the UK economy.
A quantum computer operates on a third principle again. Where a digital process is discrete and analog is continuous, a quantum process is a superposition of all possibilities. A quantum computer typically comprises a set of components called qbits, which are linked or entangled together in such a way as to collectively processes a near-infinite superposition of intangible possibilities before they find the most likely answer. Other more exotic quantum architectures also exist, which are not bound by the "bit" model necessary to communicate with a discrete-logic controller. Although the quantum processor is surrounded by a conventional digital control system, the core processor itself is anything but digital. These devices are at a very early stage of development and to say more requires a degree in advanced ... - you know, I don't even know what you do need to know.
The human or animal brain is, broadly speaking, an analogue device in terms of its chemistry but digital in the way it encodes nerve signals as pulse trains. Nobel prize-winning physicist and mathematician Roger Penrose is among those who have argued that neural processing, especially memory, also depends on quantum phenomena.
A neural network is a technological architecture which seeks to emulate certain features of human brain activity. Neural networks are commonly used for AI applications involving big data and deep learning. Typically, at some point a digital device will be used to simulate an analogue function (which is itself an approximation of the original neural digital pulse train). Theoretically an analogue device known as a memristor would greatly improve circuit efficiency, but only experimental memristor-like circuit modules have yet been developed. Working systems remain all-digital emulations.