Some people might say I have weeds in my lawn. My opinion is that I have companion plants. That is to say, rather than having a monoculture of just grass, I have a biodiverse lawn with a variety of different plants .... which leads to a nice-looking, healthy, green lawn.
Besides grass, here are several other plants that live in my lawn and keep it interesting:
Clover: Clover is the ideal companion plant for lawns. It harbors nitrogen-fixing bacteria, keeping the yard rich in nitrogen so that fertilizer is unnecessary. It also is soft on feet, doesn't need mowing, and stays green in all but the most severe drought.
Dandelion: I actually had to hunt all over the yard for a dandelion -- I guess they are being crowded out by all the other plants! Dandelion breaks up hard subsoil with its long, tough taproots. This helps make nutrients deeper down in the soil available to the shallower-rooted grasses.
Plantain: Plantain is invaluable in our climate. We have a big problem with bare patches here in Virginia. Once the soft clay soil is denuded of grass -- say, because of high traffic or erosion -- it bakes into a compacted, bricky pavement, which won't let anything sprout. I've seen a lot of attempted solutions, from just seeding it and leaving it to piling it all with straw mulch and seeding it, and nothing really works all that well. Grass just won't put a root through that compacted stuff. You know what will, though? Plantain. It easily sprouts in compacted soil, loosening and shading the soil to make it hospitable to other plants. The bare patches that covered half our front and side lawn when we bought the house are almost completely gone now, mostly thanks to plantain.
Purslane: Like dandelion, it brings up water and nutrients from below the soil. Its succulence makes it extremely drought-tolerant, and it makes a humid microclimate for the plants around it. In other words, it's happy to share its water stores with others!
A lawn of plain grass easily browns, requiring fertilizer, frequent watering, and weeding. But a diverse lawn full of "companion plants" (that is to say, weeds) stays healthy longer. Don't believe me? We're still mowing our lawn, despite intense heat and drought!