The full block chain size is currently roughly 145GB, I can handle such a download in a laptop or a desktop pc but how does it work on a mobile? I cannot download such an amount of data into my phone, so I guess there is an intermediary when using the mobile app?
There are two types of wallets:
A full node client uses the peer-to-peer network to validate transactions and blocks and to relay information to other peers. To verify all transactions, the entire blockchain (~145G) acts as a dependency to enable the tracing of transactions on the network.
SPV, Simplified Payment Verification, is used as a light-weight implementation of Bitcoin only validating your transactions by asking for data from full node peers. We use SPV clients for convenience to prevent accumulating about 145G when validating only one address worth of transactions.
Does anyone need to download the full blockchain to use Bitcoin on a mobile phone?
No. The wallet software you are talking about is a Full Node and is, by design, not available on a mobile phone due to obvious reasons. Mobile phones have light-weight implementations of Bitcoin (SPV Wallets) and therefore only ask full node peers for related data.
There is a full node for Android. It is a wrapped version of Bitcoin Core maintained by the Greenaddress founder. You can find it here: Android Bitcoin Core
If you run it in blocks-only and pruned mode, you can get full node security with significantly less bandwidth and storage footprint.
As others have noted in their answers, mobile clients for web wallets and SPV wallets have different trust models than full nodes. Web wallets usually have a custodial model, where your mobile device is simply another way to access your account. SPV wallets can perform limited verification themselves, e.g. that transactions are valid or block headers are valid, but trust their full node peers that they are receiving information about the actual chaintip and not being lied to by omission or misled about the chaintip.