Whatever happens on the blockchain is, in most cases, completely public. There are several curious things you can glean from blockchain explorers, which are available for Bitcoin and some altcoins.
A blockchain explorer is a human-readable record of transactions and the activity of miners, and it reveals several vital statistics of a network.
Your Own Address
You may explore your transactions and balances if you know your public address. For Ethereum token owners, this is the best way to preview token balances without unlocking your wallet.
But for coins like Bitcoin and Litecoin, just checking balances and being aware of transactions is a plus.
Your Last Transaction
You can track your latest transaction and its status if you paste the number into a blockchain explorer. Sometimes, you would need to track your transaction and see what slowed it down.
For mined coins, you can observe if the coin is mined as intended, with the usual relative time between blocks. Mining too fast or too slow are anomalies and not a good sign. Blocks coming in too fast means too many miners are working and the network is inefficient. A long block time means not enough miners are working at the current difficulty rate.
Tracking the number of blocks is important as usually forks and updates are announced through a block number.
You can do a quick check on how the network handles transactions by looking at the mempool size. This is a measure of the backlog of transactions waiting to be included in a block. When mempools grow, this means the network is clogged and your transaction may have to wait more. Also, mining fees would increase.
If you want to explore a new coin, look at the hash rate. If it is growing, along with the difficulty, it means miners are interested in the coin. The hash rate and difficulty for the leading cryptocoins has been on an upward trend in the past couple of years.
For most coins, you can see the distribution within the biggest wallets. This will help you estimate whether the coin can be affected by whales, or large holders. Usually, the biggest wallets belong to exchanges, but may also belong to individuals.
Mining Pool Influence
You can preview whether a coin is mined fairly. A large pool mining more than 51% of the network is worrisome, and while there have been no reports of a 51% attack, it is still better that the hashing power and block discovery are more decentralized.
Mining pool influence has been important when watching the upcoming Bitcoin hard fork, as the influence of miners may swing the decision to fork the network, depending on support.
Separate contracts and other address-related activities
With the Etherscan tool, you can do much more- track smart contracts and individual tokens. This is useful when exploring an ICO and checking if this is the right address. Comments on Etherscan may also give you an idea if you are sending to the right wallet.
The Most Useful Blockchain Explorers
CoinDance: at a glance, Bitcoin and Bitcoin Cash. Also includes market price information.
Etherscan.IO: overlook the whole Ethereum network and check your balances without opening a wallet.
Blockchain.Info: one of the oldest blockchain explorers, also gives access to a wallet. A superb tool for tracking transactions.
BlockCyper: combines Bitcoin, Litecoin, Dogecoin and a test net.
Monero blockchain explorer: while the participants are anonymous, you can still track the transactions.
CryptoID: aggregates many blockchains for various coins.
Blockchains are a tool underlining the transparency of cryptocurrencies- all activity is in the open, all processes are visible, despite the fact that individual transactions remain almost unreadable to a human viewer. But knowing your way around a blockchain explorer is the best way to take responsibility and learn how to be "your own banker".