As I am cleaning my data for my dissertation for ANCOVA, do I just need to delete responses that did not complete the entire survey? (I used a survey, and some respondents did not complete all items on the survey)
I think that you can delete incomplete surveys and you explain the exact details of the deletions in the methods and materials. But better draft such a proposed procedure first and run it by your chairman and by some of your committee members before proceeding.
Your research defines your population; in this case it would be 'people who, ..., and who completed my survey.'
It is generally not recommended to use listwise deletion of cases with missing scores as this would lead to unbiased results only under the very restrictive assumption of missing completely at random (MCAR) data. Even when this assumption is met, listwise deletion often leads to a substantial loss of statistical power due to the reduction in the sample size. Modern approaches to missing data analysis are multiple imputation and full information maximum likelihood, which only require the weaker assumption of missing at random (MAR) data and help you retain as much statistical power as possible.
It is also important for you to try to find out whether there is a reason that some participants only provided partial data to see what the relevant missing data mechanism is in your case. That is, are there any variables in your data set that are correlated with missingness? See Enders (2010), Applied missing data analysis, Guilford Press, for a very accessible treatment of missing data issues and proper ways to handle missing data.