All cookies are client and server
There is no difference. A regular cookie can be set server side or client side. The 'classic' cookie will be sent back with each request. A cookie that is set by the server, will be sent to the client in a response. The server only sends the cookie when it is explicitly set or changed, while the client sends the cookie on each request.
But essentially it's the same cookie.
But, behavior can change
A cookie is basically a name=value pair, but after the value can be a bunch of semi-colon separated attributes that affect the behavior of the cookie if it is so implemented by the client (or server).
Those attributes can be about lifetime, context and various security settings.
HTTP-only (is not server-only)
One of those attributes can be set by a server to indicate that it's an HTTP-only cookie. This means that the cookie is still sent back and forth, but it won't be available in JavaScript. Do note, though, that the cookie is still there! It's only a built in protection in the browser, but if somebody would use a ridiculously old browser like IE5, or some custom client, they can actually read the cookie!
So it seems like there are 'server cookies', but there are actually not. Those cookies are still sent to the client. On the client there is no way to prevent a cookie from being sent to the server.
Alternatives to achieve 'only-ness'
If you want to store a value only on the server, or only on the client, then you'd need some other kind of storage, like a file or database on the server, or Local Storage on the client.