Government Financial Balance: Obviously, when negative we have a budget deficit, if positive we have a budget surplus.
Current Account Balance: The current account is the sum of the balance of trade (exports - imports of goods and services), net factor income (such as interest and dividends) and net transfer payments (such as foreign aid). In the US its largest component is the trade balance (exports - imports). For example, if we import $50 of goods and export $30 of goods our current account balance is in deficit by $20 (assuming other current account components have zero balances). See here for more.
Capital Account Balance (aka Financial Account Balance): The capital account is the change in the foreign ownership of domestic assets minus the change in domestic ownership of foreign assets. In the US it is indeed mostly the financial account (which deals with only financial asset net changes). It is essentially the negative of the current account balance. In our example, the foreign sector as a whole now has a $20 surplus, with which they can buy dollar denominated assets such as US Treasuries or US buildings. So there will be a net change in foreign ownership of US assets by $20. In reality though the current account and the capital account balances don't exactly equal due to differences in accounting conventions and exchange rate fluctuations. See here for more.
Now let us take a look at some examples using the sectoral balances equation. Real world data fits pretty close to the predicted equation in every case.
Private Financial Balance = Government Deficit + Current Account Balance
If we look at data for 2009. The private balance is in surplus or the private sector is net saving, as the large government deficit more than offsets the current account deficit. This would be an addition to private sector net financial wealth by $1244 billion or 8.9% of GDP.
1243.7B = 1621.7B + -377.4B (as % of GDP: 8.9 = 11.6 - 2.7 )
If we look at data for 2006. The private balance is in deficit or the private sector is net borrowing, as the government deficit was not big enough to offset the large current account deficit. This would be a subtraction to private sector net financial wealth by $509 billion, 3.8% of GDP.
-508.9B = 291.6 + -798.4B (as % of GDP: -3.8 = 2.1 - 5.9)
If we look at data for 2000. The private balance is once again in deficit as the government is running a surplus and the current account is in deficit. Once again we have a subtraction to private sector net financial wealth by $557 billion, 5.6% of GDP.
-557.3B = -146.6B + -410.4B (as % of GDP: -5.6 = -1.5 - 4.1)
This data shows why the federal government should never run a budget surplus unless it is offset by a current account surplus.
1. See Line 37 here for private balance. Note that the definition here is NOT the same as in Line 3, see notes 2 .
2. Line 3 is disposable income minus final consumption expenditures, it does not include investment expenditures on physical capital. For example, household sector (a part of the private sector) final consumption expenditures is the 'C' component of GDP. New housing purchases are not included in 'C', they are counted under investment 'I'. So Line 3 minus net physical capital formation (such as new houses) due to investment spending will give us the private financial balance.
3. See Line 40 here for government balance
4. Current, capital, financial accounts have different meanings in different contexts in national accounting. Here we are talking about foreign transaction current account, foreign transactions capital account etc:
5. See line 29 here for US current account balance