The Standard Salesforce model
In the standard Salesforce world, the Account is paramount. An Account represents a business entity that the organization (you) wishes to do business. Contacts are merely the people who work at the Account and facilitate that business relationship.
Check out this presentation for an excellent overview of the Starter Pack and the account/contact model. Below is a slide from that presentation which visualizes the standard Salesforce relationship between opportunity (financial transactions) and accounts and contacts.
Nonprofit Starter Pack data model options
What's important to understand is that there is no way to change the way Salesforce fundamentally works. The Account will always be the center of Salesforce in connection to donations and financial transactions.
If your organization receives its funds from other organizations, this is the standard Salesforce behavior and little customization has to happen for those records.
For dealing with financial transactions from individuals and non-business donors, the Nonprofit Starter Pack is code and customization that allow a contact and household to have as strong a role in the relationship to financial transactions as the account does in standard Salesforce. The Nonprofit Starter Pack also allows for a hybrid situation where an organization receives financial contributions from a mix of individuals and businesses.
For individuals, this happens either by creating a mirror Account record with the same name as the contact (the One-to-One model), as shown below. In this scenario, code is enforced which keeps the contact front and center and relegates the account to a behind-the-scenes role. Every contact must still have a corresponding Account record, but that Account record is purely in the background.
Notice that in standard Salesforce, the Opportunity (donation or gift record) is connected directly to the Account, and the Account is connected to the contacts. There is little direct connection between the financial transaction and the individual who signed the check except as through that Account record.
In the Nonprofit Starter Pack's 1-to-1 model, the Opportunity is connected to both the Account and directly to the Contacts and the Household.
The other possible scenario is known as the Individual Bucket Model. In this case, rather than creating a hidden account record for each individual, each individual contact shares the same Individual account. While most organizations lean towards the One-to-One model, the Individual bucket model also has its strengths. Refer to the presentation for more detail.