We should probably start clarifying what is the relationship between the two. Let's put it this way:
Agile is the mindset, the WHAT
It is not about ‘doing’ Agile but being Agile.
Scrum is the framework, the HOW
It is one of the available frameworks most used to become more Agile.
The Agile Manifesto tells us that it values the items on the left more than the items on the right:
Scrum defines specific concepts and practices, divided into the three categories of Roles, Artifacts, and Time Boxes. This aims to get the most out of uncertainty.
So the question is, how is Scrum supporting the Agile principles?
The Scrum team members work together to achieve a shared business goal. The set of roles (Product Owner, Scrum Master & Development team) and how these are 'played' / contribute to each sprint is the key to success.
Each individual within the Scrum team has a high responsibility and ownership towards the goal: from refinement of the work (backlog items), to their estimates, which are actually committed (sprint items), removing obstacles, optimising the backlog, definition of what a 'Done' item means.
To enable the interactions in a consistent manner, Scrum framework uses:
Sprint Planning - To discuss items of most value, clarify, decide which to commit to the Sprint, breakdown some tasks which enables seeing different approaches / points of view for the very same item.
Daily Sprint - For progress transparency, collaboration, ownership, clarification, obstacles and team alignment.
Review - Open to all stakeholders where the 'potentially shippable product' is transparently reviewed, demoed and new possible increments may arise. The review is the 'what', the deliverable tangible concept.
Retrospective - Internal for the Scrum team where openly the 'how' is reviewed, to determine what could be changed that might make the next imminent sprint better.
The Agile principle of Individuals and Interactions is supported by the 3 pillars of Scrum: Transparency, Inspection & Adaptation.
Scrum requires a working, finished product increment as the primary result of every sprint which the team had chosen and commit to it at the start of the sprint.
Even that the increment may not be 'shipped', the team’s job is to ensure the functionality is delivered by the end of the sprint as a 'potentially shippable product' which could go to market.
During the sprint, the team works together, cross-skilled, with continuous testing, focusing on the defined goal to be achieved.
Agile values the working product more. A quick note on documentation, avoid long winded documents no one will read, think of it with your Agile hat on: modular, iterative, easy to digest and reuse.
Agile principle of Working product is supported by the Scrum pillar of Inspection.
More often product reviews (including demos) enable more often feedback, and the frame on the event sets an open and transparent environment, so that the increment of product is being delivered early and often.
The role of Product Owner is the key here to work closely with the business to prioritise the possible future increments by value (return on investment, refinement, size, risks etc)
Collaboration is supported by the Scrum pillar of Transparency. At the beginning of the sprint you have a very open and clear prioritised committed items into the Sprint and the Review openly show what was accomplished and a place to highlight obstacles that may have disturbed the flow within that cycle.
Scrum is based on empiricism (inspect and adapt) which uses feedback loops to cope with complexity and risk. Scrum emphasises decision making from real-world results rather than speculation.
Empiricism is a theory that states that knowledge comes only or primarily from sensory experience.
As new information is discovered, the team updates the backlog, adapts towards the goal and constantly responds to change to deliver what has been committed as a 'possible shippable product'.
Responding to change is supported by the Scrum pillar of Adaptation. Out of the 12 Agile principles, personally if I had to choose one it will be “Reflect regularly, tune & adjust to become more effective”, Retrospectives are the key to this. Only the organisations following this will keep up with the future.