By and large, the people on the recieving end of a blow job prefer their partner to swallow their semen. This isn't a universal truth, of course - pretty much nothing is! But it's a solid rule of thumb in almost all cases. Plenty of blow-job-givers love the act of swallowing as much as the other side of the equation enjoys watching it, and for some - particularly submissives in BDSM-style relationships it can be a fantasy and a huge turn-on.
If your partner doesn't like the idea, though, you mustn't press the issue, there are loads of valid reasons not to want to swallow someone's ejaculate, and just not feeling into it is absolutely one of them! It's possible to learn how to enjoy it if it's something they'd like to be able to do, but there's absolutely no reason anybody should feel obliged to try if they're just not into it.
Many sexual encounters involve semen to some degree, but there are plenty of folks who have a specific fetish for it - whether for swallowing it, being covered in it or having it inside them! There are a few sexual health precautions to take, of course (if one of you is capable of becoming pregnant and would rather not be, err on the side of caution with mitigating that risk; use barrier contraception where necessary to protect against STIs; you know the drill) but if you're with someone you know and trust and you're tested frequently there's no reason not to indulge in all the cumplay you might desire.
Snowballing is when one person passes on the cum from their mouth to another, increasing the spit and the 'snowball' effect. For some of the swappers (particularly those who choose to be sexually submissive) this can be a particularly erotic activity to take part in - and the other side of the equation gets to enjoy the sight of their partners loving being covered in their semen. Some people also enjoy participating in the swapping of their own ejaculate, which can be a fun thing to experiment with if the idea appeals to you!