There are four ingredients in beer (traditionally). We will only go into a bit of detail on each of them (if you are interested, you can find a great book series that do that in my books&links section).
Water is 90% of the beer. Water quality and chemicals are essential for beer and type. The hard water (meaning the high quantity of calcium) of Dublin gives us Stout and Soft water (low amount of calcium) gives us Pilsner beers (just one example).
Not much but of course, that depends on how sensitive the beer taster is. I had the chance to brew stout and Ale (Carlsberg style) from the same water, and the water quality didn't affect them. You can treat the water, even completely filter all chemicals out, and adjust the chemical as you need/want, but we will not discuss it. You can find much more detail about it online and in the recommended book with the surprising title: water.
Malt is a general name for different types of grain, like barley and wheat (2 rows or three rows, we won't go there). Some beers use Rice and/or born as well. There is the base malt which will be most of the malt used, and malt that will give you color, taste, and even smell.
The roasted malt is primarily used in darker beers like Stout as they will add color more than fermented sugars.
The Lighter malt is used mainly in lighter beers (like wheat beer) and as the base for the darker ones. The type of malt will determine the percentages of the alcohol in the beer and the head (the white stuff at the top of the beer after you pour it into the glass), some way, the beer's clarity (it can also be filtered out).
There are two main types of yeasts:
Top yeasts - are being used in Ale. Top yeasts will "float" in the top part of the wort.
Bottom yeasts - are being used in Lager. Bottom yeasts will "sink" to the bottom part of the wort, unlike their cousins.
Some brewers use both in the same brew to get specific tastes and smells. Each of these yeasts has many kinds (called 'strains' in the professional term), and each gives other things to the beer.
The primary use for Yeast is to convert the sugars in the wort into alcohol and CO2.
In the beginning (750-1000 AD), Hops were used for their quality as natural preservers. As understanding and brew science grew, brewers learned how to use them to affect the smell and taste and not just help preserve the beer longer. The hops add aroma and flavor (bitterness) to the beer (think IPA). Hops were only added to beer in the late 18th century as a natural preservative. Hops plants need specific conditions to grow: sun and weather. That is one reason only particular parts of the world can produce them. Any change in them can change the chemical qualities of hops, which will affect the acidity and other attributes, affecting the beer's smell, taste, and aftertaste.
Brewers are adding hops early to use the natural preservative quality.
Adding it later in the brew will affect the taste (bitter) and smell.