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If you are a consumer (consuming services hosted on the cloud) then - no, it is not possible to "install could on your computer".
If you are a service provider and have the necessary software (apache and a static ip) to convert your computer into a web-server, then yes - it is fair to say that you have installed cloud on your machine that is supposed to be used by "other" users.
Actually in traditional sense, Cloud is much more related to managing your "data" and "login/logout" by using a cloud database.
-YD
The Cloud is not like a software application that is installed on a PC or laptop. The Public Cloud is a computing environment that provides compute storage and network resources on servers provided by a Cloud company (hosted at a different site). If you want to install a Private cloud on your own set of servers, then you can do that by installing Private Cloud software on your servers (at your own private data center with your own servers or co-located servers).