Run query 1st month for previous month else current month

First of all you need to realise that date data is not stored in a particular format. It is simply stored as datetime data (actually 8 bytes), and it is only when we read it out of the database does a format get applied.

Given that it should be easy to follow that you need a where clause like

where
(
datepart(dd, getdate()) = 1 --1st of month
and yourdatefield >= dateadd(mm, -1, cast(getdate() as date)) --1st of prev month
and yourdatefield < cast(getdate() as date) --1st of this month
)
or
(
datepart(dd, getdate()) <> 1 -- not 1st of month
and yourdatefield >= dateadd(dd, -datepart(dd, getdate())+1, cast(getdate() as date)) --1st of this month
and yourdatefield < getdate() --now
)

I would have approached it a bit different. I would have applied the logic to the dates in variables before moving to the actual statement i.e.

DECLARE @StartDate DATE,   @EndDate DATE

IF DATEPART(dd, GETDATE()) = 1 --1st of month
BEGIN    
    SET @StartDate = DATEADD(mm, -1, GETDATE()) --1st of prev month
    SET @EndDate = GETDATE()
END
ELSE
BEGIN
    SET @StartDate = DATEADD(MONTH, DATEDIFF(MONTH, 0, GETDATE()) , 0) --1st of this month
    SET @EndDate = DATEADD(dd, 1, GETDATE())
END

Then an easy WHERE clause would have been

datefield >= StartDate AND datefield < EndDate