The definition of an aspect is 'a particular part or feature of a matter'. The relevance and significance of this to EFT is when we are tapping on an issue (particularly when we are beginning with EFT) we forget to look for or don't even see the various different parts or features of an issue and get frustrated that EFT is not working for us.
For example, if you have a charge on an issue and you rate your anger at 7 out of 10 and after 2 rounds the anger is 0 but you feel sad instead you may think EFT has not worked for you. What you have in fact done is shifted aspects, the emotion of sadness is a new part of the issue. Shifting emotions is a very important, and sometimes subtle part of learning and applying EFT.
Another example of aspects is a specific event where someone said something hurtful to you. You tap on your anger and hurt and after a few rounds it is 0. Then you remember the look on their face, and up the charge goes again and you're steaming mad, and you think "I thought EFT worked?!". You have shifted to another part or feature of the issue so you tap on how you feel about 'the look' and keep going until you have found all the aspects and there is no charge left on anything related to that specific event. This detective work will come naturally the more you tap and gain experience. So the next time you think EFT has not worked, take another look to see if it is in fact another part to the issue and tap on it. You can find out more about aspects here.