Begin by examining the alien object. Try to determine how the pattern in the center of the object can translate to the word above it: GREETINGS.
Each letter has a pattern of length four made up of dots and blanks below it. Look at other details on the alien object to determine how each pattern maps to each letter in GREETINGS. The brackets on the left and right will help.
The G in GREETINGS has the pattern: small dot, blank, blank, small dot below it. Using the key at the top, you can see that for any given left-pointing bracket, a small dot translates to going up and a blank corresponds to going down. So, the G's pattern corresponds to up, down, down, up - if you follow the series of left brackets from inside to out, you'll see that this ends up at the letter G. This holds for all of the other letters in GREETINGS. You now know how the pattern works! The next step is to translate the alien message.
Each light in the alien message turns on and off over the course of the four days. Notice how this is similar to the pattern of four you had below each letter in GREETINGS.
Each light that turns on and off over the four days represents a letter. Looking at the light in the top left of the overlaid grid, for instance, shows: small dot, small dot, blank, blank across the four days. Using the brackets on the alien object, this pattern translates to the letter D. What does the rest of the message say?
In their first communication, the aliens are asking a question of utmost import: DOES PINEAPPLE BELONG ON PIZZA?
Hope you had fun!