I'm a recent convert to the type where you load the fruit between two halves of a metal mold and squeeze.
I find that it extracts a very high percentage of the juice with minimal effort and does a good job of separating away the seeds.
I find it a lot less messy than the reamer or other type I've owned in the past.
cooking.stackexchange.com/users/1393/Michael Natkin
I use one of two things: my hands or a lemon reamer .
The lemon reamer is great if you really need to get all of the juice out of the lemon.
It takes much less effort to use compared to your hands alone.
This makes a difference if you are needing to juice several lemons; your hands might get tired.
In practice, I think I use my hands 80-90% of the time.
It's not common that I'm juicing several lemons all at once.
I don't have any experience with the more complex (expensive) tools, but I don't think there's much justification in purchasing anything fancier than a reamer.
I don't see a benefit to having a bulkier, more expensive unitasker in my kitchen.
cooking.stackexchange.com/users/60/hobodave
You should only start using one if you have a pressing need .
That is tongue-in-cheek, but unless your have a need, there is no 'should use'.
One such need may be hygiene (or saving effort, or lack of strength in the hands), but only you can determine whether that's true.
UPDATE:
original post was modified (and better for it)
As in the earlier answer, benefits may be: Hygiene Economics (more juice from your lemon, if normal squeezing isn't enough for you) Speed / efficiency Catches the seeds for you
A possibly disadvantage may be that it mangles your lemon skin (depending on type), which you might still want to use.
On the other hand, models may also exist that allow you to squeeze efficiently after you've scraped some of the skin (if you needed to), which can be a hassle if done by hand.
As to specific models: I don't use any.
(probably just use the orange juicer if I had a bunch.
Like hobodave - don't need another singletask device that sees uncommon use).
cooking.stackexchange.com/users/8457/Tobias Op Den Brouw
My lemon squeezer has a lid with the squeezer and a container underneath.
On the pro side, it holds enough liquid to make a couple of glasses of orange juice; it will separate the juice from pips and most of the flesh.
On the down side, it has two components which means more to clean, more things to search for while your cooking; this particular model has a lid that fastens too tightly.
It's easy to get covered in juice as you try and pull it off.
cooking.stackexchange.com/users/1549/Chris Steinbach
I'm a huge fan of the vintage glass bowl/reamer combination.
They're large and have a built-in bowl for catching juice, very easy to use, and extremely easy to clean.
I get great leverage standing partially over them
(I'm also 6'2" so that might factor in)
and I've never had any other tool get so much juice out, though I've never tried the type of simple hand reamer hobodave mentions in his post.
As a bonus, if you have a china cabinet or anything similar, they're pretty attractive and can be displayed if you so desire :)
My model is two pieces, there is a glass reamer that sits inside a glass bowl, I find this catches pulp way better than the single-piece models.
The top piece looks exactly like this , and the bottom (bowl) is very similar to this , except no reamer attached of course.
edit: It looks like what I have is very similar to what Chris has, except glass.
His bullets apply to mine as well, except the last con - this doesn't have any seal whatsoever, which I thought was going to make it dangerous to use, because I expected the top piece to slide as I moved back and forth.
However, between the small "point" on the bottom of the top piece, and the downward pressure
I exert keeps it from moving at all.
cooking.stackexchange.com/users/1236/stephennmcdonald
If you have relatively strong hands, I don't think it's very useful.
Just roll the lemon (or lime) on the counter, pressing down on it to break down the fruit a bit, then slice in half and squeeze each half.
A little fine-mesh sieve with a handle is great for catching the seeds, or your fingers will work in a pinch.
Quick and one less item to store and wash!
On the other hand, larger citrus can't really be juiced that way, and not everyone has a handshake of steel, so some people might well want a gadget.
cooking.stackexchange.com/users/45/Harlan
One benefit could be saving your hands and skin from citrus infection.
I never let lemon juice or any other strong acid get on my hands, it messes with the skin and can cause citrus infections (which is a common problem for bartenders cutting and squeezing limes and lemons by hand).
cooking.stackexchange.com/users/615/Tore
There are  lots of products  on the market to ease lemon squeezing. I've never used any of them.   Are there any benefits to using one? What are some of the advantages and disadvantages of the different types?
Are there benefits to using a lemon squeezer?
cooking.stackexchange.com
<equipment><lemon>
cooking.stackexchange.com/questions/4854
cooking.stackexchange.com/users/1799/Mehper C. Palavuzlar
A traditional lemon reamer with a glass bowl is good for getting all of the juice out of the lemon, but other options include squeezing between two halves of a metal mould or squeezers with a container underneath. Many basic systems exist from a lemon reamer to a squeezer with a built-in container to catch the juice.
Squeezing the fruit in a metal mold is a good system. A lemon reamer is very efficient. A lemon squeezer with a lid and a container underneath is a good option.