Jeff, piercing the veil is a legal term that involves a determination by a court that the corporate entity can be ignored and the individual held personally liable for what would otherwise be corporate debts. As a high level overview, it generally happens only when the owners of the corporation ignore the corporate formalities and run the business like it is their own personal business.
The check deposit activity that Luis describes is one aspect of the types of actions a court examines. Others include failure to maintain corporate formalities such as holding board and shareholder/member meetings, commingling of funds, paying personal expenses with corporate funds, etc.
A single member LLC is not inherently more likely to have the veil pierced. On the other hand, the owner of a single member LLC may be more likely to fail to follow the corporate formalities, and may be more likely to engage in behavior that a court would consider as evidence that the veil should be pierced, because that person is not accountable to anyone else. For example, if a major corporate decision requires the approval of the members, a multiple member LLC may be more likely to actually hold a meeting to discuss the subject and prepare formal minutes documenting the decisions that were made at the meeting. The single member LLC owner may just make the decision in his or her mind and then act accordingly, with no record of the action.
So, a single member LLC is fine, as long as you follow the rules and truly treat it as a separate legal entity.
Ralph, I know it seems silly to hold a meeting with yourself, talk to yourself, and agree with yourself on something (sounds like some of those people we see standing on street corners all day), but that is essentially how the documentation should appear. Indeed, you should really have an occasional written document that says the member of the LLC has determined that the LLC should do x, and sign it as member.
Ideally, you would have similar documentation to what you would expect with a multi-member LLC. At a minimum, there should be an annual meeting, just like corporations are required to have an annual meeting of shareholders.
Your LLC agreement may specify what actions require approval of the member(s). Otherwise, think about what you would expect to have agreed in writing if there were multiple members, e.g. decisions about banking, major hires, investment, obviously M&A.