SQL Server cluster node fail over details
Just to add, your sql agent job could look like this, which works for us (this assumes you have database mail configured properly):
Set the schedule to be when sql server starts as described in Ronaldo's answer
DECLARE @importance AS VARCHAR(6)
DECLARE @body AS NVARCHAR(1000)
DECLARE @Subject  AS NVARCHAR(1000)
DECLARE @InstanceName AS VARCHAR(100)
DECLARE @NodeName AS NVARCHAR(100)
DECLARE @recipientsList VARCHAR(100)

SELECT  @recipientsList ='someuser@mycompany.com;someuser@mycompany.com'
SELECT  @InstanceName =@@SERVERNAME
SELECT  @NodeName = CAST(SERVERPROPERTY('ComputerNamePhysicalNetBIOS') AS NVARCHAR(200))
SELECT  @Subject  = 'Failover occured for SQL Server Instance '+@InstanceName
SELECT  @body = 'Failover occured for SQL Server Instance '+@InstanceName + 'This instance is currently running on the node '+@NodeName
SELECT  @importance ='High' 
EXEC msdb.dbo.sp_send_dbmail 
     @profile_name ='Dbas',
     @recipients=@recipientsList,
     @subject = @subject ,
     @body = @body,
     @body_format = 'HTML' ,
     @importance = 'HIGH'