There's enough breast pump choices on the market now to baffle any new mom. There's manual vs. electric, single vs. double, closed vs. open system, hospital grade vs. personal use, renting vs. purchasing and if you manage to get through all those choices, then there's choosing the right brand and accessories.  This guide will clarify what all these differences are, and what some of the accessories are.

1. Manual vs. Electric

There is a world of difference between these.  A manual pump is one completely powered by your hands.  Manual is fine for occasional or on the fly pumping (I once used a manual to pump while my mom was driving!) but not ideal for frequent pumping and can take a long time if you have to do both sides, since you can't do both sides at the same time with a manual pump, unless you have 3 hands. 
Electric has a motor of some sort that does the actual pumping for you. You just hold the horn(s) -cone that goes on the breast- and control the speed/suction.  Electric is the ONLY way to go for frequent daily pumping, and mandatory for you if you need to build a supply up.

2. Single vs. Double

Manual pumps come with a single pump only, since you can't do both at the same time when pumping manually.  Some electric pumps have the option of being purchased singly as well.  A single pump is fine for on-the-go or infrequent pumping, but inconvenient and time consuming for frequent pumping.  If you pump more than once a day, do not get a single pump for your main pump. I used to keep a single manual pump in the diaper bag because I had such severe supply problems and had to be stimulated every hour to build up, but other than that, the single was terribly time consuming.  A double makes much more sense. I mean, if you're going to sit there anyways, get double the results while you're doing it! You will have both hands busy holding horns to your breasts, but you will be done faster than if you have to first do one side and then do the other.  A double electric pump is the most efficient one, ideal for a working mom or mom who will be pumping milk and then feeding by bottle for some reason.

3. Closed vs. Open System

This refers to whether the pumping system (only applicable with electric pumps that have tubing going from the motor to the horn that goes on your breast) has the ability to get flooded with milk and therefore unsafe for another user.  This has recently become an issue of worry.  An open system (Medela and most others) means that it is possible, however unlikely, that while you are pumping you could overflow the collection container, and the milk would go up through the tubing and get into the motor. Since the motor can not be sterilized, the milk would remain there and ferment or mold.  That could be dangerous to your own child, or, there is a chance that if that pump is then resold (or you buy a used pump) a baby could get sick from something mom#1 had in her milk.  They haven't documented a case of this happening yet, they're just saying it could.  A closed system (Ameda) means that there's no way for milk to get into the motor and so no need to worry about the above problem.  Hospital grade pumps escape this worry. They are made especially for multiple users, though each user has to have a brand new collection kit.

4. Hospital Grade vs. Personal Use

Hospital grade is a must for any mom with supply problems. If you have polycystic ovarian syndrome, diabetes, or any other health condition, ask your Dr. if it will affect your supply.  They are more powerful and able to fine-tuned to fit you better and so are better at quickly building a milk supply.  They are also the best for pumping every few hours every day, like if you have  preemie who can't nurse yet but you want to give him breastmilk.  They are widely available for rent, and also available for expensive purchase. Lots of moms initially rent a hospital grade pump to build supply, and then purchase a personal use pump for the duration that they nurse. Personal use is a step down from hospital grade, intentioned for only one user (though you CAN buy them used on eBay and rent them at some locations).  The most common Personal Use is Medela Pump In Style.

5.  Renting vs. Purchasing

Hospital grade pumps are best to rent, as their purchase cost is inhibitive.  But if you're going to nurse what's recommended by the American Academy of Pediatrics or beyond, purchasing a personal use pump makes a lot of sense. Most pumps (hospital and personal) rent for $30-$70 monthly, so spending $300 or less on a pump you get to keep and use with more children comes out cheaper than renting for 12 months or more.  I think it is best to rent a hospital grade and have it waiting for you when you get home from the hospital. Use it for a month at least to build your supply, and then purchase (or have already purchased) a personal use pump for when you send the rental back. This way you're ready for whatever may happen.

6. Brands

Nowadays pretty much every brand that makes baby stuff makes a pump. But not all pumps are created equal, and Medela still leads the pack for quality and ease of use.  They are easy to find, easy to use, and come with lots of accessories to make breastfeeding easier.  You get what you pay for. If you see a double electric pump for $50, it's not going to do the job you need it to do.

7. Accessories

This category could go on and on with the stuff they've come up with to make pumping easier.  One required accessory for most double electric pumps is a collection kit. Even if you rent one, it will not come with the tubing needed to use it. At the top of the list for convenience in breastfeeding is the "Pumping Free" attachment for Medela nursing bras. It holds a pump in place at each breast so you don't have to hold them there the whole time. They don't always work perfectly, but it can give you a break sometimes. Milk storage bags are essential, and buy generic, since usually they'll fit any nurser bottle. Medela sells breast shields in varying sizes and textures so you can find the perfect fit.

So now you know what everything is. You have to decide WHICH is right for you.