Since government borrows the money in term of balancing budget and will have to repay using budget, shouldn't it make more sense to define the ratio against something related to the size of the budget, instead of the size of the whole economy? Someone pointed out to me that when seen this way, the debt ratio sounds more serious. For example, US public debt is ~18 trillion, which is slightly more than the annual GDP (this doesn't sound too bad), but actually roughly 6x government revenue in 2014 (~3 billion according to CIA World Factbook)
The ratio of government debt to GDP is a useful indicator because, broadly, the larger a country's GDP, the larger is the tax revenue the government could potentially raise to service the debt while maintaining other government expenditure at an acceptable level. Thus a very high debt / GDP ratio would raise doubts about the government's ability to service the debt.
One reason why a larger GDP implies a larger potential tax revenue is that most of the transactions making up GDP from an income perspective, eg wages and salaries, company profits, could be taxed. However, taxes could also be levied on items that are not part of GDP, eg capital values of property. The more fundamental point, therefore, is that a larger GDP, again from an income perspective, tends to imply a larger aggregate ability to pay, out of that income, such taxes as may be levied by the government.
The government's ability to tax is also influenced by political factors and by the effectiveness of its tax collection, so the debt / GDP ratio can only be a crude indicator of ability to service debt.
I believe your understanding of budget and GDP is a bit off. Comparing debt with GDP is more adequate, as GDP measures the country's production level, while budget is just an estimate. At least where I live, what is budgeted for the upcoming year is not always accomplished in the end, as there are gaps and things that missed the planning phase.
In that case, it makes a bit more sense to compare what the country owes (debt) with what it is actually producing, rather than with what it estimates to produce, don't you think? I'm not exactly sure, but I believe GDP aggregates indirect revenues, while revenue size does not - hence looking more serious.