There are still a lot of misconceptions out about SPV wallets, people make it seem like they are less useful than they are, and this is a shame.
SPV wallets are wallets with “Simplified Payment Verification”. A term that actually shows up in the original Bitcoin whitepaper from Satoshi Nakamoto.
The trick is that SPV wallets (which in practice is almost all wallets) check all the Bitcoin blocks, validate if the miners did their work but they do not download and check all the transactions. This limits the download and check of the entire 8 year blockchain to about 39 MB. Pretty neat eh?
So, what are people worried about?