How does a national budget differ from a household budget?
If you think about it, there is really not much difference between a government budget constraint and a household budget constraint. 

Both have uncertain income streams, which are labor and capital income for the household, and mostly tax revenue for the government. On the other side of the constraint, you will find consumption and saving for the household. The government will spend most of its income on (public) goods and transfers to households (social system).

Both can move income inter temporally using borrowing. However, neither can run a permanent deficit where it is "not expected to ever pay back", something that is typically referred to as a No-Ponzi-Game condition. As an implication, the "life-time budget constraint" will have to hold for both:

sum of all expected future income = sum of all expected future expenditures


I'm pretty informal here, you would typically rather derive "present value" of these constraints, but latex is not yet allowed here, and I'm already writing much.

What should government debt be used for?

We typically assume that households and governments have concave utility, they prefer consumption processes that are second-order dominating. Hence, they want to engage in smoothing of consumption over fluctuations.

Since all debt has to be paid back eventually, there is limited use of government borrowing. Most typically, one wants to stabilize consumption and output over business cycles, by financing consumption in busts and paying back in booms. Also, given an interest rate r, investments that promise a higher return than r can be financed through debt, as the government will not violate consumption smoothing.

Potential Difference

One potential difference is the commitment. Developed governments are typically more 
trustworthy, they are more likely to pay back there debt. Hence, they can borrow at cheaper interest than most households. As a conclusion, they can use debt more aggressively than households for the two purposes consumption smoothing and investment, as emphasized above.

tldr: Besides that, they're very much alike. Note also the Ricardian Equivalence, which is not at the heart of this question, but very much related.