Can I use regression to analyze relationship between rating and choose-all-that-apply data? If I sent all of my customers a product to try, let's say it is a laundry detergent product. I then ask them to rank their liking of this product, from 1 to 9. then I ask them 'which words do they associate this product with?' The answers could be: fresh scent, convenient, clean, etc. (this is choose all that apply question). 
Can I use regression to find out which 'descriptor' contributes to the overall liking of this product?
 A: It depends on whether you are looking to identify the relationship between the overall rating of the product (it's not a ranking, but a rating) and each descriptor separately, or trying to determine if combined, how predictive those words are of the overall rating.
In your case, a customer could select more than one word, so you would need to create separate binary explanatory/independent variables for each word, indicating whether that word was selected by a customer (1 = selected, 0 = not selected).  If you then create a multiple regression model, using all of those explanatory variables together, you will not be able to estimate the individual contribution of each variable to the overall rating.  This is because, in a multiple regression, the estimated coefficient for each variable will need to be interpreted as "the average difference in rating between someone who selected that particular word and someone who did not select it, if they both had the same selections (or lack of selections) for all other words."  In a multiple regression, any correlations between different explanatory variables can affect both the statistical significance and magnitude of the estimated coefficient on each variable, so you would not be able to determine how much each descriptor contributes to the overall rating.
So if you are interested in identifying the relationship between the overall rating and each descriptor separately, not accounting for any correlation between selections of different descriptors, you should not use a multiple regression.  Instead, given that you have an ordinal dependent variable (rating data is considered ordinal type data) and categorical independent variables (with two categories: Yes/No), you can perform something like a Kruskall-Wallis test (which tests the null hypothesis that the means of ratings in each group are the same).  You could also create categories of rating values (e.g., rating of 5 or more as one category, rating of 4 or less as the other category), and perform a Chi-square test.
If you are going to proceed with a multiple regression model (or even do another simpler statistical test), you should do some more reading on how to properly model this type of dependent variable (e.g., if and when it is appropriate to treat Likert/rating data as interval type data vs. ordinal data, and what kinds of assumptions you need to check).  Ordered logistic (or probit) regression is recommended for modeling ordinal dependent variables, but it comes with a "proportional odds assumption", which says that the coefficients that describe the relationship between, say, the lowest versus all higher ratings are the same as those that describe the relationship between the next lowest rating and all higher ones.  This assumption should be tested.
You can perform a search on this site like this: https://stats.stackexchange.com/search?tab=votes&q=likert%20regression, which will return results such as this answer: https://stats.stackexchange.com/a/1887, or this: https://stats.stackexchange.com/search?tab=relevance&q=ordinal%20regression.

If you have not already done so, before starting any statistical analysis, I would strongly encourage you to first plot rating values for people who selected and who did not select a particular word, as that might already give you some indication of how the overall rating varies across groups.  Furthermore, performing statistical analysis on this kind of data may lead to misleading results.  In an extreme hypothetical example, if the group of customers who selected "clean" all either rated the product as 1 or 9, while customers who did not select that word all rated the product as 5, any statistical test that analyzes differences in means will return an insignificant difference between the two groups.  Or it is possible that a group that selected/did not select a particular word is very small, making it more likely that its average rating is an extreme value, and that kind of issue also needs to be accounted for in statistical analysis.
A: Yes you can. 
Remember you will need a large number of datapoints to make this a reasonable investigation.
I would recommend an ordered choice model.
When modelling these types of outcomes, numerical values are assigned to the outcomes, but the numerical values are ordinal and reflect only the ranking of the outcomes.
There may be a natural ordering as in your example, the OLS model would not be suitable and an ordered probit choice model would be useful. 
You can then look at the marginal effects of the explanatory variables and see which ones are significant etc. 
