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ActivateDate ShipDate Month Month_Length Day-2c Day-2c_Length YEAR-201x SHIPDateConcatenate ActivateDateConcatenate
NULL 6/12/2018 12:00:00 AM 6 1 12 2 2018 6-12-2018 NULL
NULL 6/12/2018 12:00:00 AM 6 1 12 2 2018 6-12-2018 NULL
NULL 6/12/2018 12:00:00 AM 6 1 12 2 2018 6-12-2018 NULL
NULL 6/12/2018 12:00:00 AM 6 1 12 2 2018 6-12-2018 NULL
NULL 6/12/2018 12:00:00 AM 6 1 12 2 2018 6-12-2018 NULL
NULL 6/12/2018 12:00:00 AM 6 1 12 2 2018 6-12-2018 NULL
NULL 6/12/2018 12:00:00 AM 6 1 12 2 2018 6-12-2018 NULL
10/12/2018 14:45 10/16/2018 12:00:00 AM 10 2 16 2 2018 10-16-2018 10-12-2018

The two columns [ActivateDate] and [ShipDate] datatype is listed below; however each time I try to use convert() or cast() to a date type, a conversion error occurs.

SELECT
 [ActivateDate], -- '10/12/2018 14:45' nvarchar(100)
 [ShipDate], -- '6/12/2018 12:00:00 AM' nvarchar(100)
 SUBSTRING(iedimpr.[ShipDate],CHARINDEX('/', iedimpr.[ShipDate])-2,2) as 'Month',
 LEN(SUBSTRING(iedimpr.[ShipDate],CHARINDEX('/', iedimpr.[ShipDate])-2,2)) as 'Month_Length',
 REPLACE(SUBSTRING(iedimpr.[ShipDate],CHARINDEX('/', iedimpr.[ShipDate])+1,2),'/','') as 'Day-2c',
 LEN(REPLACE(SUBSTRING(iedimpr.[ShipDate],CHARINDEX('/', iedimpr.[ShipDate])+1,2),'/','')) as 'Day-2c_Length',
 SUBSTRING(iedimpr.[ShipDate],CHARINDEX('/201', iedimpr.[ShipDate])+1,4) as 'YEAR-201x',
 SUBSTRING(iedimpr.[ShipDate],CHARINDEX('/', iedimpr.[ShipDate])-2,2)
+'-'+REPLACE(SUBSTRING(iedimpr.[ShipDate],CHARINDEX('/', iedimpr.[ShipDate])+1,2),'/','') 
+'-'+SUBSTRING(iedimpr.[ShipDate],CHARINDEX('/201', iedimpr.[ShipDate])+1,4) as 'SHIPDateConcatenate',
 SUBSTRING(iedimpr.[ActivateDate],CHARINDEX('/', iedimpr.[ActivateDate])-2,2)
+'-'+REPLACE(SUBSTRING(iedimpr.[ActivateDate],CHARINDEX('/', iedimpr.[ActivateDate])+1,2),'/','') 
+'-'+SUBSTRING(iedimpr.[ActivateDate],CHARINDEX('/201', iedimpr.[ActivateDate])+1,4) as 'ActivateDateConcatenate'

After reading the question again I've noticed the sample data is ill-formatted, so I've edited it. During this edit I found that you are using american style (mm/dd/yyyy) for your string representation of dates.

To convert a valid string representation of DateTime with this format into date, you need to use 101 as the style parameter in the convert method.

However, I highly recommend using Try_convert instead of Convert, since it will simply return null when the value can't be converted instead of raising an error.

That being said, here is an example:

First, create and populate sample data(Please save us this step in your future questions):

DECLARE @T AS TABLE
(
 ActivateDate nvarchar(100),
 ShipDate nvarchar(100)
)

INSERT INTO @T(ActivateDate, ShipDate) VALUES
(NULL, '6/12/2018 12:00:00 AM'),
('10/12/2018 14:45', '10/16/2018 12:00:00 AM'),
('20/14/2018 14:45', '10/16/2018 12:00:00 AM'), -- invalid ActivateDate
('2/4/2018 14:45', '10/16/ZOIB 12:00:00 AM') -- invalid ShipDate

The query:

SELECT ActivateDate, 
 ShipDate,
 TRY_CONVERT(datetime, ActivateDate, 101) As DateActiveDate,
 TRY_CONVERT(datetime, ShipDate, 101) As DateShipDate
FROM @T

Results:

ActivateDate ShipDate DateActiveDate DateShipDate
NULL 6/12/2018 12:00:00 AM NULL 12.06.2018 00:00:00
10/12/2018 14:45 10/16/2018 12:00:00 AM 12.10.2018 14:45:00 16.10.2018 00:00:00
20/14/2018 14:45 10/16/2018 12:00:00 AM NULL 16.10.2018 00:00:00
2/4/2018 14:45 10/16/ZOIB 12:00:00 AM 04.02.2018 14:45:00 NULL

If you only need Date (without the time part), simply use try_convert(date, ...) instead of try_convert(datetime, ...).