Well, the definitions are ambiguous, nobody can agree on them anyway. I was working from the context of the rest of his post:
from what I have gathered there are Impact Driven Algorithms,Cost
Driven Algorithms,Opportunistic Algorithms,Other(Multi leg,Volatility
Driven,GWAP).
These are all strictly execution algorithms.
Where do Mean Reversion and Momentum strategies belong?
These are aspects of possible Black Box trading algorithms, or hybrids.
Can a simple household use algorithmic trading or do they rely
brokerage firm to execute their strategies?
A simple household will almost certainly rely on a brokerage firm to execute their strategies; they will not be using a home-made algorithm to jockey for queue position nor will they implement their own VWAP algorithm.
Is algorithmic trading dependent on Direct Market Access(DMA) or is it
not necessary?
Algorithmic trading, assuming we are talking about algorithmic execution, will be severely handicapped without Direct Market Access, but it's not impossible. It is, however, practically impossible on a platform like Quantopian, where you have no event-driven opportunities to change or cancel orders. Algorithmic investing, quantitative/automated trading, these are a different kettle of fish, this stuff is definitely possible without DMA, and Quantopian is proof enough of that.
Are there limits to what you can trade through algorithmic trading?
It's difficult to write automated execution algorithms for assets for which there is not an electronic market. OTC derivatives, for instance.
I'm using Barry Johnson's Algorithmic Trading and DMA book.
Excellent books, I'd say the best, for algorithmic execution.
I am interested in Algorithmic Trading and not High Frequency
Trading/BlackBox
Great, because that's what I have been assuming! Black Box trading is also known as automated trading, or quantitative trading, or algorithmic investing. It is characterized by a completely automated process from trading signal generation to position management. It might not involve any custom algorithmic execution, because, as you say, it's difficult for simple households and retail investors to implement fast-enough event-driven strategies. Typically, people rely on their brokers to execute their orders. High-frequency trading is just Black Box trading except very fast, where other challenges come into play, such as hardware latency, physical location, the speed of light and so forth.