See: Tier 1 Capital. See: Tier 1 Capital Ratio.
In general, tier 1 capital provides a buffer for banks in times of trouble. The term applies to the financial institution's really reliable assets...the ones it knows will be there in an emergency.
How much tier 1 capital a bank needs in part gets determined by the size of the bank. A bank with $1 million of tier 1 capital would be doing extremely well if it had $4 million in total assets. However, a bank with $1 million in tier 1 capital would be in real trouble if it had $100 million in assets.
For that reason, regulators use tier 1 capital ratios to determine a bank's financial health. The most common measure, called simply the "tier 1 capital ratio," takes the amount of tier 1 capital a company has and compares it with the risk-weighted assets that it holds. That is, it measuers the tier 1 holdings against the assets the bank holds, as judged in accordance with the amount of risk they carry. The assets are ranked by how likely they are to pay off. Those risk assessments are baked into the calculation.
The tier 1 leverage ratio uses a different measure of total assets. Instead of risk-weighted assets, it looks at total assets, without the risk assessment. It provides a different way to look at the same question. The underlying issue still relates to the bank's ability to withstand troubled times. The leverage ratio just compares the tier 1 capital with a different measure of the bank's total capitalization.