Server Object Model?
SharePoint server where the Server Object Model binaries

Example

inside Web Parts / Workflows.

Client Object Model?

client machines where the entire SharePoint binaries are not available.

Example
Windows Forms application in a client machine.

"Client" in the namespace.  For example, in the server object model you have:
Microsoft.SharePoint.SPSite
Microsoft.SharePoint.SPWeb
Microsoft.SharePoint.SPList

In the client object model you have:
Microsoft.SharePoint.Client.Site
Microsoft.SharePoint.Client.Web
Microsoft.SharePoint.Client.List


NameSpace:


Microsoft.SharePoint.Client.dll
Microsoft.SharePoint.Client.Runtime.dll


Example Code:

ClientContext context = new ClientContext("http://peakfinders.com");

Web web = context.Web;

List list = web.Lists.GetByTitle("yourlistname");
ListItemCollection listItems = list.GetItems(new CamlQuery() { ViewXml = "<View/>" });
context.Load(listItems);
context.ExecuteQuery();

listItems[listItems.Count - 1].DeleteObject();
context.ExecuteQuery();

"Client" in the namespace.  For example, in the server object model you have:
Microsoft.SharePoint.SPSite
Microsoft.SharePoint.SPWeb
Microsoft.SharePoint.SPList

In the client object model you have:
Microsoft.SharePoint.Client.Site
Microsoft.SharePoint.Client.Web
Microsoft.SharePoint.Client.List


NameSpace:


Microsoft.SharePoint.Client.dll
Microsoft.SharePoint.Client.Runtime.dll


Example Code:

ClientContext context = new ClientContext("http://peakfinders.com");

Web web = context.Web;

List list = web.Lists.GetByTitle("yourlistname");

CamlQuery query = new CamlQuery();
query.ViewXml = "<View/>";

ListItemCollection listItems = list.GetItems(query);

context.Load(listItems);

context.ExecuteQuery();

ListItem item = listItems[listItems.Count - 1];
item["Status"] = "Complete";
item.Update();

context.ExecuteQuery();

"Client" in the namespace.  For example, in the server object model you have:
Microsoft.SharePoint.SPSite
Microsoft.SharePoint.SPWeb
Microsoft.SharePoint.SPList

In the client object model you have:
Microsoft.SharePoint.Client.Site
Microsoft.SharePoint.Client.Web
Microsoft.SharePoint.Client.List


NameSpace:


Microsoft.SharePoint.Client.dll
Microsoft.SharePoint.Client.Runtime.dll


Example Code:

ClientContext context = new ClientContext("http://peakfinders.com");

Web web = context.Web;

List list = web.Lists.GetByTitle("yourlistname");

ListItemCreationInformation newItem = new ListItemCreationInformation();
ListItem listItem = list.AddItem(newItem);
listItem["Title"] = "Peakfinder";
listItem.Update();

context.ExecuteQuery();