What is an Execution context and what things to keep in mind when developing on the platform
Let's say you are logged in to salesforce and you want to create an account. You fill in all the necessary fields for accounts (Name is the only required field for Account) and an Account gets created.
The processing that happens when you hit the create button, to the account being created in the system can be considered a transaction.

Now, there can be multiple operations happening when this account gets created. Like field validations, triggers are fired on account creation, which my lead to more such operations, roll-up summaries calculations, class methods, you get the point

You can imagine the scale at which these things are happening when there are multiple users in the system doing multiple transactions.

Why should you care? Limits
Its important to understand the concept of a transaction because salesforce limits are tied to each of these transaction.
If you dont know, limits are way of shuting down the usage of the system resources by salesforce such that one transactions does not use up a lot of system resources and clog up the processes.So to keep thing free flowing salesforce imposes limits on each transactions. Like SOQL 101 , DML etc.

Transaction vs Execution context
As per Apex developer guide, it is one and the same so don't get confused when you see these terms being used.

Static variables
Unlike other programming languages, A static variable is only available in the transaction boundary. A static variable cannot be used to maintain state across transactions. There are other ways to maintain state which we will not be discussing here.
Database commits and Transactions
To keep the database in a consistent state, there is an all-or-nothing nature to the DMLs fired in a transaction. What it means is if an account is inserted, related to that an opportunity is inserted, and related to that an OpportunityLineItem is inserted and if the system hits any kind of exception or error, all the operations are rolled back and nothing gets inserted.

