What are the differences between algorithmic trading, automated trading and high frequency trading?
What is automated trading?

Automated trading is a very general term of involving a computer program that creates orders and trade automatically. However, in practice, it means automating trades that humans were already doing manually.

What is algorithmic trading?

Algorithmic Trading refers to a computer science driven approach to automated trading where the trading strategy is being researched and improved.

Borrowing from some of the answers below:

Automated trading is a very general term of involving a computer program that creates orders and trade automatically.
Algorithmic trading can be described as automated trading using computers, which are predefined to do certain responses correspond to the movement of the market.
High-frequency trading is one type of algorithmic trading, which involves a large number of volume of shares are being bought or sold automatically with a very high speed.
By @MarcoWong
and

Algorithmic Trading refers to using strategies which are executed in line with some underlying logic or rules based system designed to allow optimal execution of trades. Typically using short term signals in order to do so.
High Frequency Trading (HFT) involves algorithmic trading on much shorter timescales, where signals are processed and orders are executed within microseconds. In this domain - speed of action and low latency are key to exploiting structural inefficiencies in the micro-structure and "plumbing" of the markets and exchanges to generate returns.
Automated trading involves automating the decision making and execution of trades. It includes both HFT and Algorithmic Trading.
by @BharatRao
Risk Parity and Trend following approaches to Asset Allocation use simple rules to determine appropriate weights for Asset Classes using information which is already available to the portfolio manager such as historical volatility in the former case, or historical prices in the latter. As such, given predetermined rules, machine learning techniques for prediction are not needed.

In the Algorithmic Trading space, machine learning techniques can be applied to processing large amounts of data in order to perform factor selection. ML inspired reduced form models have been shown empirically to perform better at equity premium prediction than more rigid structural models that adhere to economic theory.
Machine Learning / A.I. models work very well when the relationships are not very clear. However when using ML models the practitioner faces a problem, that of parameter overfitting. While strategies like Trend Zfollowing are already being recast into ML-driven time-series indicators based trading, strategies like Risk Parity have tried to reduce as many parameters as possible.
Different strategies can be combined ideally to generate a continuous and diversified signal to trade upon, this can be achieved via considering strategy correlation and the different alpha vectors generated and optimising the combination of these to generate a more robust and general signal to trade upon. Such signals may be combined conditionally and non-linearly through optimisation and machine-learning processes.