The term cloud is bandied about with gay abandon without much definition. Specifically it can mean many things but in general terms it means putting thins to other than your local PC or device. This means that someone somewhere has computers that can perform tasks remotely for you which you can access via the internet. This may be as simple as storing things for you, take Dropbox for instance. In other cases it may be a suite of applications, such as Google Drive which includes apps such as editing and spreadsheets. Google is an excellent example since it encompasses a suite of useful functions such as mail, calendar, Apps and storage.The "cloud" is now so ubiquitous that every major IT company now offers some form of cloud to their customers/users, some more accessible to your regular Joe such as us and others aimed higher at the small to medium business and others more suited to large companies.
The disadvantages are that someone else has your data, good solutions can be expensive, you are relying in the viability of the company hosting your data and applications, you are relying on theis stability and security to no have your data compromised or lost. There have been a lot of instances of data lost, services failing and passwords compromised so this is a serious consideration.