In short: Kiva is a charity that facilitates microfinance throughout the world.
Kiva itself is a 501(c)(3) charity with the stated mission "to connect people through lending to alleviate poverty." (Kiva - Abous Us).
There are a lot of ideas about how to help solve the poverty problem through the world. The old proverb about teaching a man to fish makes a lot of sense, but if the man can't get a fishing pole or net then he's still going to be hungry. The loans on Kiva help him buy what he needs to catch his own fish.
I live in the United States. Like many people in the first world, I've taken for granted the ability to borrow money at reasonable rates. If I need to borrow a few hundred or thousands of dollars, I can go to dozens of local banks and credit unions and probably get that money with an annualized interest rate on the order of 5-15%, or maybe upwards of 20% if I'm a bad lending risk. This ability to borrow facilitates small business, which makes up a huge portion of our economy (55% of US jobs, according to the Small Business Administration.
If you live in Kenya or Turkmenistan, your borrowing options are much more limited, though. Big capitalists are often hesitant to loan to individuals in developing nations because they represent much higher risk. Without this backing capital, small loans at reasonable rates aren't possible.
What is a reasonable rate? According to "Banks Making Big Profits From Tiny Loans" (The New York Times), the global average is about 37% with some countries averaging as high as 70%, and some institutions charging as high as 125% APR. For a sense of scale, a $500 loan for one year at 15% APR costs $75 in interest, while at 70% it costs $350. I personally define anything less than the global average of 35% as reasonable, for this purpose. I try to loan only through institutions with a portfolio yield or average interest rate below 30%. Komak is one of my favorite institutions, as they appear to only charge enough interest to cover their losses and have a high repayment rate.
Kiva is a middleman to provide backing capital for microloans around the world. They have over 200 partner institutions that actually provide loans to people and a team of volunteers that go on site to try to verify that these institutions are charging reasonable rates and not conducting any bad business practices.
These field partners take loan applications and decide whether or not to give out the loans like any other bank does. When the approve a loan, they give the money (or buy the goods directly, in some cases) and post the loan to Kiva to raise that money from people like us. When you search the list of loans and buy a $25 share in one, that money goes to backfill the loan.
As payments are made to the field partner, your share of it is deposited back to your Kiva account. The interest that is charged goes to the field partner, so they're making all the profit on your lending capital. This lets the field partner grow and be able to facilitate more loans in the future, hopefully helping to establish a better market for credit in the area.
One of Kiva's goals is to promote more transparency in lending. To that end, they have quite a few statistics available on their field partners that allow you to make informed decisions about your lending. I have found some institutions that have an average interest rate that I feel is too high, so I simply don't back their loans.
I like to think of microloans as charitable lending. The best case scenario is that you get 100% of your money back — earning 0% profit. Various factors can lead to loan defaults or currency losses, so your actual rate of return will probably be a bit less in the long run. That loss is your charible donation. The rest is capital that you can reinvest or withdraw back into your bank account.
I've made it my personal goal to make at least one loan per month for all of 2014, though I've been averaging more like 2 or 3. So far, I've gotten all of my projected repayments on time, which I promptly reinvested into another loan. I also give about $2 to Kiva for each loan I make, to help them cover their expenses. While this is less than their suggestion, I feel like it should be enough. That donation is tax-deductible.
You can see the loans I've made on my lender page, as well as a list of the teams to which I belong and statistics about who and where my loans have been to. If you're interested in joining, you can sign up with this link to give me credit for referring you. When you do that, they let me lend out $25 of their money as a bonus.