You can use the provider-specific ConnectionStringBuilder class (within the appropriate namespace), or System.Data.Common.DbConnectionStringBuilder to abstract the connection string object if you need to. You'd need to know the provider-specific keywords used to designate the information you're looking for, but for a SQL Server example you could do either of these two things:
System.Data.SqlClient.SqlConnectionStringBuilder builder = new System.Data.SqlClient.SqlConnectionStringBuilder(connectionString);
string server = builder.DataSource;
string database = builder.InitialCatalog;
or
System.Data.Common.DbConnectionStringBuilder builder = new System.Data.Common.DbConnectionStringBuilder();
builder.ConnectionString = connectionString;
string server = builder["Data Source"] as string;
string database = builder["Initial Catalog"] as string;