Subplots are like mini-stories which are interwoven with your main story. They can involve your main characters having two things going on in their lives at the same time (e.g. finding love in the course of solving a mystery), or secondary characters having their own mini-story.
And no, you don’t absolutely need one. If your main plot is rich, intriguing and solid enough to carry the whole story, then you can get away without one.
It can be a very good idea to include one, for a number of reasons:
It adds depth to the story. Done well, it makes the story more multi-layered, and somehow more satisfying to the reader for that.
If your subplot is about secondary characters, it can make them more rounded, as the reader sees them have their own challenges too.
It can make it easier on you as the writer. It can be hard to sustain one story for 100,000-or-so words. Having two stories going on spreads the load, so to speak.
It can help build tension. If you leave the main plot on a cliff-hanger and go to the secondary story, it’ll keep the reader wondering and in a state of curiosity and tension. (You need to do this well though, lest it become irritating and annoying.)
If the main story and the secondary story are intertwined (see below), the secondary story can help with the outcome of the main story.
Subplots can be parallel, or interwoven.
Parallel stories can often be the easiest to do. They tend to involve characters who know each other - that’s the link which ties them together in the story. But the plots are completely independent of each other.
Often the characters can come together for some purpose - book clubs was a favourite device over the past few years, and I’ve seen holiday resorts used the same way. Originally the characters have nothing in common. But being at the same book club or holiday resort gives them something, no matter how tenuous, in common. The story would then follow each character as his or her situation is explored, and somehow resolved.
Done badly, this can seem only a clumsy device to tell a few people's stories. Done well, the experience they share can help them resolve their personal challenges, and each can learn from the others and be influenced by them.
Interwoven plots are much more complex. In interwoven plots, the outcome of the main plot depends in some way on the outcome of the subplot. Say the subplot was about (to pick a mad example off the top of my head) the secondary character’s fear of water and their attempts to overcome it.
In this case, perhaps the denouement of the main plot would depend somehow on this secondary character swimming through water. Wanting to help the main character would provide motivation and incentive for the secondary character to finally conquer their fear of water and thus provide a nice ending to the subplot. But in turn that resolution would directly contribute to the resolution of the main plot as the hero (in our example) was rescued from the water.
As discussed elsewhere, this would be better if it’s not too obvious. What we’re aiming for is that the reader will get a surprise at the ending, but at the same time realise that it was so obvious an ending, so right, and hence so satisfying. Not easy, not always accomplished, but definitely worth aiming for.
Another example of an intertwined plot was done very well in Recipes For A Perfect Marriage by Kate Kerrigan. The main character was having difficulties in her marriage and found her grandmother’s diary. Through the diary we, the readers, became fully immersed in the grandmother’s story too, and her marriage. And at the end, the outcome of the grandmother’s marriage gave the main character an understanding of her own marriage, and contributed to the resolution of that plot. (A lovely book, by the way, beautifully written, and highly recommended.)
So, in summary, a subplot is a very useful tool, and it’s well worth considering including one in your story.