Try this:
DECLARE @tmp TABLE([Name] VARCHAR(30), [Date] DATETIME, Holidayname VARCHAR(30) NULL)
INSERT INTO @tmp ([Name], [Date], [Holidayname])
SELECT 'mohammed.shareef' AS [Name], '6/1/2013' AS [Date], 'Weekend' AS [Holidayname]
UNION ALL SELECT 'mohammed.shareef', '6/2/2013', 'Weekend'
UNION ALL SELECT 'mohammed.shareef', '6/3/2013', NULL
UNION ALL SELECT 'mohammed.shareef', '6/3/2013', 'Holiday'
UNION ALL SELECT 'mohammed.shareef', '6/4/2013', NULL
UNION ALL SELECT 'mohammed.shareef', '6/5/2013', NULL
UNION ALL SELECT 'nagarjuna.doddipatla', '6/1/2013', 'Weekend'
UNION ALL SELECT 'nagarjuna.doddipatla', '6/2/2013', 'Weekend'
UNION ALL SELECT 'nagarjuna.doddipatla', '6/3/2013', 'Holiday'
UNION ALL SELECT 'nagarjuna.doddipatla', '6/4/2013', NULL
UNION ALL SELECT 'nagarjuna.doddipatla', '6/5/2013', NULL
SELECT DISTINCT t1.[Name], t1.[Date], (SELECT [HolidayName] FROM @tmp WHERE [Name] = t1.[Name] AND [Date] = t1.[Date] AND [HolidayName] IS NOT NULL) AS [HolidayName]
FROM @tmp AS t1
ORDER BY [Name], [Date]
Warning! This is not optimal solution (time of execution would be long for thousands of data).