I can't save Database Diagrams
Hard to suggest a more specific fix or suggest a cause given that we have no background on how your system got into this state, what you did prior to trying to create the first diagram, if you have successfully created diagrams in the past, or even what version of SQL Server you're using. It sounds like your sysdiagrams table somehow lost the IDENTITY property on the diagram_id column. Do this:

DROP TABLE dbo.sysdiagrams;
GO
CREATE TABLE [dbo].[sysdiagrams]
(
    [name] [sysname] NOT NULL,
    [principal_id] [int] NOT NULL,
    [diagram_id] [int] IDENTITY(1,1) PRIMARY KEY,
    [version] [int] NULL,
    [definition] [varbinary](max) NULL,
    CONSTRAINT [UK_principal_name] UNIQUE ([principal_id],[name])
);

GO
EXEC sys.sp_addextendedproperty 
  @name=N'microsoft_database_tools_support', 
  @value=1 , 
  @level0type=N'SCHEMA',
  @level0name=N'dbo', 
  @level1type=N'TABLE',
  @level1name=N'sysdiagrams';
GO


This will of course lose any diagrams you were previously able to save.

(After performing a search it saddened me to see that many people suggest as a fix to make the default for this column 0. This is great after you create one diagram but falls apart for multiple  reasons once you try to create a second one.)