The 'President's House', built in stages between 1792 and 1829, is an iconic, imposing building that's thrilling to see, but difficult to access. Tours must be pre-arranged: Americans must apply via one of their state's members of Congress; non-Americans must ask their country's embassy in DC for assistance – in reality, there's only a slim chance that the embassy will be able to help source tickets. If you're lucky enough to visit on a tour, you'll see several rooms in the main residence.
Applications are taken from 21 days to three months in advance; the earlier you request during this time frame the better. If this all sounds like too much work, pop into the impressive White House Visitor Center a block away. It's not the real deal, but it does a good job in recounting the property's history and evoking life in the presidential residence.
Cars aren't allowed to pass the building on Pennsylvania Ave, so it's possible to take good photos across the North Lawn from this vantage point. Or move to E St NW and take pictures across the South Lawn.