How should I choose the name of my past degree in application when the options do not not exactly match my degree? I am applying for PhD for a certain university (Canadian), and in the form there is a question about the academic background. In the options I see things like Bachelor of engineering, bachelor of engineering with honours, bachelor's degree general, bachelor's degree 4 year, and bachelor of science. There are similar options for the Master's degree: Master of engineering, Master's degree, and master's of science.

I completed my undergraduate study in 4 years and my degree is "Bachelor of science in engineering".There was a project in the final year. I am doing a "Master's of science in engineering". This is thesis based degree.

Which of the above options should I choose? In my country we only have Bachelor of science and Master's of science so I don't know what these things mean.
By "I see things like," I assume you mean "the options are," i.e., the list of options you have given is exhaustive. I shall also assume that you are an engineering student, with a degree at "Bachelor of science" level and you are currently studying at "Master's of science" level. The question about academic background is looking for your highest qualification, so I would list "master's of science."

You could call the university to confirm!

EDIT. I do not agree with compass's remark that a UK MEng degree is provided in lieu of a Bachelor's and has the same study length: UK BEng degrees are typically three years, whereas UK MEng degrees are typically four years. It is possible that compass is attempting to compare Bachelor's degrees between different countries.