An Initial Coin Offering (ICO) and an Initial Public Offering (IPO) are both ways for projects to publicly raise funds.
IPOs tend to be for more established and mature companies. There are far stricter rules around IPOs and stronger accountability.
ICOs have been gaining a lot of traction in the crypto space. They are usually executed by a small team with a good idea that they would like to get off the ground. They are essentially a form of crowdfunding, similar to Kickstarter, and appeal to the same audience. Regulations surrounding ICOs are still being developed, so there is much higher risk investing in one.
A team looking to raise money through an ICO would usually start by producing a white paper outlining their idea and business strategy, although they are not required to by law. A project then raises funds by selling their own cryptocurrency tokens. Organisations wanting to raise funds through an ICO would use a platform, such as Ethereum, to create their cryptocurrency token. They would also write the token rules, for example the total supply, how they will be distributed, and other special conditions.
Once the token is created, the platform serves as a register for all transactions that take place in the future.
A few companies that have done ICOs include Telegram, Block.one and EOS. If you're interested in investing in a crypto project through an ICO, make sure you do your own extensive research first.