meaning of "full node" in Lightning Network? According to this answer: (Link to answer),


  "A 'full node' refers to a fully validating node on the
  Bitcoin network, meaning the node validates each block and transaction that happens on the network. 
  
  A 'lightning node' refers to a node on the Lightning Network,
  which is a network that exists on top of the bitcoin network."


There is a list of Lightning Network mobile wallet applications here: (Link to list) in which there is a column as "Requires own full-node ?" such that in case of some wallet applications, it is answered by "Yes".

What does this requirement mean exactly?
The lightning node secures the channel funds in form of Bitcoin transactions. 

When a lightning node negotiates an initial balance with another Lightning peer (channel opening), it needs to ensure a confirmed funding transaction, which is broadcast and confirmed in the Bitcoin blockchain. When the channel is closed, the closing transaction needs to be confirmed in Bitcoin.

The lightning network represents a network of payment channels between 2 peers respectively. The funds of each of these channels, however, are secured by signed Bitcoin transactions, and the only way to confirm that the channel has been opened (confirmation of funding tx) or closed (confirmation of closing tx) is to validate these in the Bitcoin strong chain. 

Payment channels represent a transaction "layering" technique, by which transaction chains are not all confirmed in the Blockchain, except during the opening and closing of the channel lifetime. Throughout the lifetime of the channel, each peer must also remain vigilant to potential cheating of the counterparty. This cheating manifests itself as a confirmed transaction, only observable by the counterparty on the Bitcoin strong-chain.

The only way to validate the strong-chain, is by operating a full-validating node. That is why all lightning nodes require a trusted or local full-node. If a lightning wallet does not require a local full-node, it is using a trusted public one instead, or operates a SPV light client, which does not perform full-validation of the blockchain, but solely follows the chain with most PoW (a security trade-off for lower validation costs).