Dwarfs is my largest army and the one I have played the most out of all my armies in 7th and 8th ed, but not by much since I play a lot beastmen too. I got into Ogres for two reasons, first I liked the models and had been collecting them slowly over the years, and second because both armies to make the move properly would require me to paint nearly a 100 models. I have 10,000 of dwarfs, and as someone others have, said there are many ways to play them, but most of the time now days dwarfs use great weapons. I am sadly short on great weapons.
I did a post in the Beastmen portion and I will do this one like wise talking about common units and why you will run into them, and then finish with common setup concepts.
Warriors and Long Beards (warrior type)
these guys should be the bulk the force you fight, the many reason you will probably see less Long Beards than in the past is the extra cost for little gain per model. A good reason you will see them, and something to keep in mind is that Long Beards can have Rune Standards with up to 3 magic runes that total less than 50 points. I commonly put the Rune of Courage +30 and the Rune of Determination +20 (Immune to Fear and Terror and One use only make a leadership test on 1d6) on my Long Beards. Another Rune I put on there is the Master Rune of Grungii +50 (5+ ward save vs ranged weapons and magic missiles)
BTW Dwarfs do not have volley, that is purely for bows only, so Crossbows, throwing axes, and guns need not apply as far as I know.
For ranged units there are basically thunders, which are awesome need 3s and 4s at long to hit, and are str4 armor piercing at 24 inch range. They can have shields and I know some run them that way since they are the same stat line as warriors, just more costly and in light armor rather than heavy. Quellers on the other hand have picked up some traction in 8th ed. They can use great weapons, which is huge for dwarfs, but neither group can have magical banners.
Two more quick notes, Rangers are a single choice unless they have Bugman a named Hero, and because dwarfs only have a 3 inch movement base they tend to charge more often than shoot if you get too close to them and less likely charge you if you are just within short range for them.
Were the only unit of dwarfs that got a 2+ armor save in 7th ed, but now the rules are different and they have a 3+. They also can have a standard with up to 50 points of runes, and something to consider is that a dwarf can double up on non-master runes as long as the way the runes are, are different from other rune combos. So with these guys I normally put a BSB in there with the rune that give +1 CR to the unit called the Rune of Battle +25, and the same rune plus the Rune of Determination +20 (the test leadership on 1d6 as before). IB have a 4/4 s/t and are WS 5, so they are pretty elite troops, but they cost a lot too because of it. Also if you are running WYSWYG they cost a lot of real life money to buy, so you tend to see them in smaller numbers.
These almost always have a Dwarf Lord in them because he gives the unit bonuses for him being there. While he is there the unit is fearless. The unit is always Stubborn, so they are hard to move, they also come with Great Weapons, and can buy shields. Someone said they have armor, that is partly true, they do have a 5+ armor save, but against even a str 4 bull with armor piercing the hammers will not get any saves.
Slayers are harder to deal with imo, they can use 1 great weapon or 2 hand weapons, with the pair of weapons their strength raises to match the toughness of the person they are fighting maxing at 6. This means that they can hurt your badasses. They are also unbreakable, and have better initiative than other dwarfs.
War Machines are harder to deal with with dwarfs, but they can start to cost a lot if people buy a lot of upgrades for them. One thing to keep in mind is that an anvil is a war machine, so engineers can give bonuses to it so it count as defending terrain. So it might take slightly more to kill than at first thought. Also all war machines are stubborn.
Gyrocopters are more of an issue than many think at first in the new rules. They use a low str flame template and fly around quickly.
Oathstones are huge in 8th ed, but there are some rules to know about them. There can not be any other hero or lord in the unit other than the oathstone bearer. It gives him and the unit magic resist 1, he must challenge or accept challenges with the bearer. Once it is placed the unit can never move unless it flees. However, the unit can be flanked, hit in the rear, but they are not disrupted and they can make supporting attacks in any direction. If the unit does flee for any reason the bearer gives up his vp as if he had died. There is no limit to how many you can have, but you can not have it with shield bearers.
Shield Bearers are a general choice that must be a dwarf lord. it gives him two extra attacks at WS5, S4, I2, and it gives him a 1+ armor save which can not be improved. I personally ran this with the Master Rune of Steel +50 (all incoming attacks greater than strength 5 is treated as strength 5), and I think it is called the Rune of Resistance +25 which allows you (to reroll all failed armor saves). So at the very worst a Dwarf Lord with this combo could have a 4+ armor save with rerolls, and a 4+ talisman ward save for 45 more points out of their 125 allowed. The last thing I would do is get the Rune of Speed +5 on my weapon so I had a magic weapon that gave me +1 Initiative.
Anvils are used only by Rune Lords, and can have up to 150 points in runes. This means they can be quite offensive, or extremely defensive against magic or melee. Common items now taken steal one power dice and give it to the dwarfs in your phase. With a combination of items the dwarfs can get up to a +3 defense against spells, and due to the lag of how runes work they can have up to 6 normal dispel scrolls and 6 more special dispel scrolls that eats the spell as well on a 4+ if points allow that many hero and lord choices.
It is very common that a dwarf player will have more dispel dice than you have casting dice, and they have nothing to do in their turn but dispel any RiP spells in play on their turn. I would never expect to have any spell last long even if it is not able to be dispelled.
Most gun lines still have 2 blocks of melee 25 models or more with nearly max artillery with 30 to 50 ranged troops. It is the same with all of them too, if it goes off in their favor, regardless of your army you will have more dead or fleeing than moving forward. I have shot to a man a Warriors of Chaos army by end of turn 2 at 2250 with a tournament list. That is not normal though, and I could not fail a roll... my two Organ Guns wounded 34 out of 40 hits for example. Also gun lines normally run an anvil so they can slow key units.
Large melee armies suffer from other armies because of greater mobility (While Ogres are faster, they are bulky and really do not count as being more mobile imo). However, many or all of the units will have great weapons, and will probably rune dwarf lord or a rune lord with an anvil so they can get flank charges with greater ease.