Pretty much, yes, with some nuance. "The cloud" isn't some abstract, placeless thing. It's real physical servers sitting in real data centers owned by companies like Amazon (AWS), Google, or Microsoft (Azure), and when your data is "in the cloud," it's stored on hardware in one of those buildings, not floating anywhere.
What "cloud computing" actually means is renting computing resources, storage, processing power, software, over the internet instead of owning and running the physical equipment yourself. Before cloud computing, a company that needed a website would buy its own servers, house them, cool them, maintain them. Now they just rent space and processing power from a provider's massive data center, paying only for what they use, and scaling up or down as needed without buying new hardware.
For regular consumers, this is why your phone photos can sync to Google Photos or iCloud: your phone uploads the data over the internet to one of these companies' servers, where it's stored on drives in a data center, often duplicated across multiple physical locations for backup, and made available back to you (or your other devices) on demand. So yes, it is sitting on someone else's computer, but that "someone else" is typically a professional data center with security, redundancy, and backups well beyond what most personal devices have.
The trade-off is that you're trusting that company with your data and depending on your internet connection to access it. If their servers go down or you lose internet access, you temporarily lose access too, that's the most common downside people run into.