Honestly, partial definitions don't really cut it, and depending on how a question is asked, the most straightforward answer to the immediate question or situation can only give you a portion of the whole.
Total Depravity, yes, it's about man's inability to chose anything other than sin on the basis of man's - you guessed it - total depravity.
It's not to say that man doesn't choose, just that he will choose sin every time when left to his own will..
There are stages to coming to God, what Catholics call ordo salutis, which means "order of salvation". And every Christian believes that there is an order spelled out in the Bible...to the reformed, election comes first (through Gods will alone) then God works and acts to bring that about.
Reformed believe that election is unconditional, meaning that it's not based in anything that God sees in us, which makes us worthy of his choosing us.
The most common objection to this thought process is that it prevents voluntary choice, and denies the need for Gospel invitation.
The above is an untrue stance -we believe we voluntarily choose, and we absolutely believe Gospel invitation is necessary.
what we believe is that our choices are free, but that God works sovereignly through our desires so that he guarantees that our choices come about as He has ordained, but this can still be understood as a real choice because God has created us and He ordains that such a choice is real. In short, we can say that God causes us to choose Christ voluntarily.
The mistaken assumption underlying this objection is that a choice must be absolutely free (libertarian freewill) (That is not in any way caused or influenced by God) in order for it to remain a genuine human choice, and we disagree with that.
We believe God can work in us and through us and it still be our voluntary choice.