What are the most important financial metrics and equations for a startup?

There are a couple that I find important:
1. DSO- Days sales outstanding on AR & AR Aging % greater than 90 days.
2. Cash Flow statements on a bi-monthly basis. A short-term and a long-term analysis.
3. Comparing actual to budget & prior year in regards to a lot of P&L items (Revenue, GM, Operating Income, etc.).
4. Revenue per FTE (For SW companies I find that $150k per FTE is breakeven and $170k and above your company should be making money.
5. Metrics on your sales staff. "Funnel" reports to monitor their customer cycles. Don't wait until 12 months from now to find out that this sales rep is not going to make it. The sales funnel should be moving along in the cycle or it shouldn't be there.

These are just a few.

In addition take a look at annual cash compensation costs (fixed base salary and annual variable cash) versus long-term compensation (equity) costs.

To properly answer the question, it would be helpful to know in which industry the start up is or planned to be. There are different sets of metrics depending on whether the business will be product based or services based.

However, in either case, performance against a cash budget is critical. In almost all start up situations, cash flow is negative at the beginning and has to be compared to cash available from all sources (owner's cash, investor cash, credit lines and term loans).

Then from a longer range view, is the company on track to become cash flow positive in "x" months/years?

I agree with Henry, dependent on the industry, some of the KPI's will change. Generally unless you are backed by a Fortune 100 company, the cash burn rate is the most critical measurement in a start-up. Next is your timeline for actual revenue generation - staying on top of your timeline for going live with product or services and how delays impact your burn rate and cash flow planning. Most start-ups underestimate both of these issues and end up scrambling because of it.

I like these for FINANCIAL metrics for manufacturing and retail:
Direct margin percentage.
EBITDA [earnings before interest, taxes, depreciation and amortization] divided
by invested capital
Net income plus depreciation and amortization divided by debt [current and non-
current]
For retail, add percentage of sales sold at full price ["sell through"]

For other industries, other metrics are crucial, like "net interest margin" for banks.

A few critical operating metrics are also very desirable,

I have used:

Current Ratio
A/R Turnovers/Avg Collection Period
A/P Turnovers/Avg Collection Period
Inv Turnovers/Avg Collection Period
Working Capital

I must say I agree with Regis Quirin; I'm freshly reminded of the value of cash and the need to control both what comes in / goes out. To this end, for startups, rather than waiting until cash or lack of cash is an issue, I'd highly recommend 'stringent' oversight / attention to who owes you what and restrict to the highest possible extent possible commitments to spend $. Focusing on both of these will make your cash last longer, and heighten the possibility of enterprise success.

If you are an early stage startup it’s cash and burn rate. Number of customers/users/uniques, sales closed, and how much committed in your fundraising efforts are also important.

There are different KPIs for different stages of the startup. In general, cash flow is always #1. It is also important to understand when the company will turn cash flow positive (how many months/years?). Investors are also keen on keeping a pulse on the monthly burn rate. If your product/service is out there and marketed to customers, KPIs such as number of leads, funnel analysis reports, converted sales, revenue etc. become valid.

I agree with what's been posted, particularly around keeping a close on on Cash Flow (both inflows and outflows). Most young companies don't have the credit facilities to get through prolonged cash droughts.

An additional financial metrics, if you are in the construction industry, is knowing your bond working capital. Calculate by adding cash, contract receivables excluding A/R over 90 days, retainage receivables, cost in excess of billings, less total current liabilities, then multiply the results by 15.

I co-chair the software committee for the largest investor angel investment group. We want to see a strong sales pipeline to support the inevitable hockey stick revenue projections start-ups show for fund raising. Here is a short article on "Highly Accurate Sales Forecasting" https://www.proformative.com/b2b-directory/opero-partners/whitepapers.

While I agree with the above for companies which develop and sell products , If you are in the b to c internet space you want to look at metrics such as

Monthly unique visitors and growth
Time spent on site which measures engagement
Purchase conversion rates
Average Dollar spending
Customer acquisition costs