For describing a verb in Chinese, you need to use "地" Such as what you said "妈妈高兴地笑了" "得" is used to connect verbs and adverb. For example, 跑"得"很快 or 读得很慢 or you can change to 很快"地"跑 or 很慢"地"读 I know it is difficult to distinguish "地" "得" "的" in Chinese. But for native speakers, we sometimes mix them all.

妈妈开心地笑了。 My mom laughed/smiled happily. 他大声地唱歌。 He sang loudly. 我渴得喝了五杯水。 I'm so thirsty I drank 5 glasses of water. (resulting in me drinking 5 glasses of water) 我高兴得睡不着。 I'm so happy I can't sleep. (resulting in me not being able to sleep)

妈妈高兴地笑了。我渴了，连续喝了五杯水。

地：一般组词土地、地方、地区。我有一块土地可以用作耕种。这个地方很偏僻。这个地区由县管辖。 得：获得、取得、得到。我获得诺贝爾奖。这间公司取得某个项目发展。我得到了他的原谅。