This tip is to give a brief background to the two main ways to connect from your ArcGIS Desktop to your ArcSDE Geodatabase, more detail is available at;
Two tier connection (also called direct connect)
Usually the preferred type of connection is via what is termed a two tier connection, where you connect directly from your computer to the database which will be on one of the corporate servers. The term two tier refers to the first tier being the ArcGIS Desktop application and the second being the database being accessed.
A prerequisite for this is to have the appropriate database client software installed on your computer, for example if it is an Oracle database you are connecting to you will need an appropriate 32 bit Oracle Client installed. Your IT department will be able to assist with installing and configuring this software if you do not already have it (you should already have the SQL Server Native Client software installed). There may also be some additional database configuration to ensure you are using the correct account credentials to access this geodatabase (to just have access to the data you need) and that the database can support these additional connections.
The type of database you are connecting to will determine how to fill out the Add Spatial Database Connection properties, in the case below we are connecting to SQL Server on server lap-303750 using database authentication, although your administrators may prefer you using Operating system authentication in which case they can tailor data access just for you.
The following link provides more detail;
Three tier connection
This is where there is another (application server) layer of ArcGIS software called ArcSDE on a server (in some case also on the database server) between your ArcGIS Desktop and the database. The application server (ArcSDE service) is now the second tier, and the database the third tier. This is usually slower as there is an additional service between you and the data you need.
Again the connection properties are different and the Service: name is either the name of the service or a port number like 5151.
Please note that for all connections using Database authentication that saving the username and password and sharing map documents (MXD files) which have this authentication information used within them is a potential security risk.