The Budget deficit is the amount a government has to borrow in a particular year. Total Debt (known as public sector debt or referred to as national debt is the total amount of debt that the government owes – to private sector, Central Bank e.t.c)
In the case of Greece, they already had a high debt, before the onset of the crisis. Even during reasonable growth, they had a high budget deficit and national debt as a % of GDP.
Note, if you had a budget deficit equal to 3% of GDP and economic growth of 3%, then national debt as a % of GDP would remain the same. Negative growth in this recession is another factor that has increased the debt as a % of GDP in many countries.
An international comparison of debt via Spanish Tragedy