In a nutshell
Aspirationals are defined by their love of shopping, desire for responsible consumption, and their trust in brands to act in the best interest of society 
It is interesting to observe that some consumers (and it is growing) are considering that brands should stand for something bigger than product benefits. Brands should have a positive impact on the world. Brands should contribute to build the consumer’s utopia.
This Aspirational generation isn’t defined by age, but rather the desire for their actions to meet their needs, have a positive impact on others and connect them with an ideal or community that’s bigger than themselves. 
In the Marketing Canvas
In the Marketing Canvas, we have identified 6 main categories for building your Marketing Strategy: CUSTOMERS, BRAND, VALUE PROPOSITION, JOURNEY, CONVERSATION and METRICS. Each of these categories, have 4 dimensions which means that a total of 24 dimensions (6 by 4) are defining your Marketing Strategy.
ASPIRATIONS is a one of the 4 dimensions of the CUSTOMERS category.
How to use it?
Understanding the customers of your VALUE PROPOSITION is a fundamental tenet of a robust Marketing Strategy. One of the dimension that will help you improve your understanding is the Understanding of the Aspirations of these customers? What do they aspire? What is their beliefs and utopia?
Aspirationals are happy shoppers and they see brands as badges of their own identities. They expect brands to lead with their values and meet higher standards for health, safety, fairness and social impact in ways that drive progress for people and positive impact for the world.
If you don’t understand what your potential buyers/users are expecting from Brands and not only from products or services of these Brands, you risk to be out of their conversation and buying list. Why? Because they buy for consuming AND realising their utopia. If you match only their product needs but miss their utopia you won’t probably be the chosen brand. And by the way, realising utopia has a value (see my post on economic value).
While the Job To Be Done is helping you to understand the problem people are trying to solve when buying your product/services, the Aspirations will help you to understand their utopia they would like to realise through their consumptions. Some scholars are considering that Job To Be Done could be used for both, I personally prefer to split it and use Job To Be Done only for their personal problems they are trying to solve.
Is the understanding of the ASPIRATIONS of your customers is helping you achieve your goals?
More on the Marketing Canvas