The method looks fine, the logic looks fine... everyone has given everything you need to delete a row... all that is left to double check is you. :)
All I mean by that is, sometimes we get caught up in the code when soemthing doesn't seem to be working, and fail to step back a little.
First, make sure you are running your code against the right database and table. Sometimes, people use test tables that mimick production tables, then when they are testing code they might be changing a test table, but looking at the production table expecting the result.
Second, if you are getting the field value to look for from a textbox, keep in mind that 1- the query is case-sensitive, so if the [name] field contains "ABC", it won't be found using name='abc'. 2- make sure there are no extra spaces before or after the text in the textbox, you can make sure by simply using
"delete from table where name='" + textbox1.Text.Trim() + "'"
Third, if you have a window open in your IDE showing the contents of that table, then execute the code, and don't refresh that grid showing you the rows, it could appear like the record is not gone... but it very well could be. (kind of an obvious one... but it happens...)
Lastly, let everyone know if you are getting any exceptions or messages when executing this code, or if it runs fine but just doesn't seem to be removing the record...
Best of luck!